Over the course of his life, Gottlob Frege formulated two logicalsystems in his attempts to define basic concepts of mathematics and toderive mathematical laws from the laws of logic. In his book of 1879,Begriffsschrift: eine der arithmetischen nachgebildeteFormelsprache des reinen Denkens, he developed a second-orderpredicate calculus and used it both to define interesting mathematicalconcepts and to state and prove mathematically interestingpropositions. However, in his two-volume work of 1893/1903,Grundgesetze der Arithmetik, Frege added (as an axiom) whathe thought was a logical proposition (Basic Law V) and tried to derivethe fundamental axioms and theorems of number theory from theresulting system. Unfortunately, not only did Basic Law V fail to be alogical proposition, but the resulting system proved to beinconsistent, for it was subject to Russell’s Paradox.

Until late in the 20th century, the inconsistency in Frege’sGrundgesetze overshadowed a deep theoretical accomplishmentthat can be extracted from his work. TheGrundgesetzecontains all the essential steps of a valid proof (in second-orderlogic) of the fundamental propositions of arithmetic from a singleconsistent principle. This consistent principle, known in theliterature as “Hume’s Principle”, asserts that forany concepts \(F\) and \(G\), the number of \(F\)-things is equal tothe number \(G\)-things if and only if there is a one-to-onecorrespondence between the \(F\)-things and the \(G\)-things. ThoughFrege derived Hume’s Principle from Basic Law V in theGrundgesetze, the subsequent derivations of the fundamentalpropositions of arithmetic from Hume’s Principle do notessentially require Basic Law V. So by setting aside the problematicBasic Law V and the derivation of Hume’s Principle, one canfocus on Frege’s derivations of the basic propositions ofarithmetic using Hume’s Principle as an axiom. His theoreticalaccomplishment then becomes clear: his work shows us how to prove, astheorems, the Dedekind (1888)/Peano (1889a) axioms for number theory fromHume’s Principle in second-order logic. This achievement, whichinvolves some remarkably subtle chains of definitions and logicalreasoning, has become known asFrege’s Theorem.

The principal goal of this entry is to present Frege’s Theoremin the most logically perspicuous manner,without usingFrege’s own notation. Of course, Frege’s own notation isfascinating and interesting in its own right, and one must come togrips with that notation when studying Frege’s original work.(See the entry onFrege’s logic for a precise and systematic study of the notation in Frege's formal system.)But one need not understand Frege’s notation to understandFrege’s Theorem, and so we will, for the most part, put asideFrege’s own notation and the many interpretative issues thatarise in connection with it. We strive to present Frege’sTheorem by representing the ideas and claims involved in the proof inclear and well-established modern logical notation. With a clearunderstanding of what Frege accomplished, one will be better preparedto understand Frege’s own notation and derivations, as one readsFrege’s original work (whether in German or intranslation). Moreover, our efforts below should prepare the reader tounderstand a number of scholarly books and articles in the secondaryliterature on Frege’s work, e.g., Wright 1983, Boolos 1990, andHeck 1993, 2011, and 2012.

To accomplish these goals, we presuppose only a familiarity with thefirst-order predicate calculus. We show how to extend this languageand logic to the second-order predicate calculus, and show how torepresent the ideas and claims involved in Frege’s Theorem inthis calculus. These ideas and claims all appear in Frege 1893/1903,which we refer to asGg I/II. But wesometimes also cite his book of 1879 and his book of 1884 (DieGrundlagen der Arithmetik), referring to these works asBegr andGl, respectively.

• 1. The Second-Order Predicate Calculus and Theory of Concepts
  • 1.1 The Language
  • 1.2 The Logic
  • 1.3 The Theory of Concepts
• 2. Frege’s Theory of Extensions: Basic Law V
  • 2.1 Notation for Courses-of-Values of Functions
  • 2.2 Notation for Extensions of Concepts
  • 2.3 Membership in an Extension
  • 2.4 Basic Law V for Concepts
  • 2.5 First Derivation of the Contradiction
  • 2.6 Second Derivation of the Contradiction
  • 2.7 How the Paradox is Engendered
• 3. Frege’s Analysis of Cardinal Numbers
  • 3.1 Equinumerosity
  • 3.2 Contextual Definition of ‘The Number of \(F\)s’: Hume’s Principle
  • 3.3 Explicit Definition of ‘The Number of \(F\)s’
  • 3.4 Derivation of Hume’s Principle
• 4. Frege’s Analysis of Predecessor, Ancestrals, and the Natural Numbers
  • 4.1 Predecessor
  • 4.2 The Ancestral of a RelationR
  • 4.3 The Weak Ancestral ofR
  • 4.4 The ConceptNatural Number
• 5. Frege’s Theorem
  • 5.1 Zero is a Number
  • 5.2 Zero Isn’t the Successor of Any Number
  • 5.3 No Two Numbers Have the Same Successor
  • 5.4 The Principle of Mathematical Induction
  • 5.5 Every Number Has a Successor
  • 5.6 Arithmetic
• 6. Philosophical Questions Surrounding Frege’s Theorem
  • 6.1 Frege’s Goals and Strategy in His Own Words
  • 6.2 The Basic Problem for Frege’s Strategy
  • 6.3 The Existence of Concepts
  • 6.4 The Existence of Extensions
  • 6.5 The Existence of Numbers and Truth-Values: The Julius Caesar Problem
  • 6.6 Final Observations
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. The Second-Order Predicate Calculus and Theory of Concepts

In this section, we describe the language and logic of thesecond-order predicate calculus. We then extend this calculus with theclassical comprehension principle for concepts and we introduce andexplain \(\lambda\)-notation, which allows one to turn open formulasinto complex names of concepts. Although Frege’s own logic israther different from the modern second-order predicate calculus, thelatter’s comprehension principle for concepts and\(\lambda\)-notation provide us with a logically perspicuous way ofrepresenting Frege’s Theorem. We shall sometimes remark on thedifferences between the calculus presented below and the calculus thatFrege developed, but such remarks are not intended to be a scholarlyguide to the many subtleties involved in understanding Frege’soriginal works.

1.1 The Language

The language of the second-order predicate calculus starts with thefollowing lists ofsimple terms:

• object names: \(a\), \(b\), \(\ldots\)
• object variables: \(x, y,\ldots\)
• \(n\)-place relation names: \(P^{n}, Q^{n}, \ldots \ \ \ (n\geq1)\)
• \(n\)-place relation variables: \(F^{n}, G^{n}, \ldots \ \ \(n\geq 1)\)

The object names and variables denote, or take values in, a domain ofobjects and the \(n\)-place relation names and variablesdenote, or take values in, a domain of \(n\)-place relations.Objects and relations are to be regarded as mutually exclusivedomains: no object is a relation and no relation is an object. Whengiving examples of \(n\)-place relation names or variables for \(n\geq2\), we often write \(R, S,\ldots\) instead of writing\(P^2,Q^2,\ldots\).

From these simple terms, one can define theformulas of thelanguage as follows:

• If \(\Pi\) is any \(n\)-place relation term and \(v_1,\ldots,v_n\) are any object terms \(( n\geq 1)\), then \(\Pi v_1\ldots v_n\)is an (atomic) formula.
• If \(\phi, \psi\) are any formulas, then \(\neg\phi\) and \((\phi\to\psi)\) are (molecular) formulas. (We drop the parenthesisaround \((\phi\to\psi)\) when there is no potential forambiguity.)
• Where \(\phi\) is any formula and \(\alpha\) any variable, then\(\forall\alpha\phi\) is a (quantified) formula.

So, for example, \(Pa\), \(Rxy\), etc., are atomic formulas and theseassert, respectively, that object \(a\) exemplifies the 1-placerelation \(P\) and that \(x\) and \(y\) stand in the relation \(R\).The formulas \(\neg Pa\) and \(Pa \to Rxy\) are molecular formulas,and these assert, respectively, that it isnot the case that\(a\) exemplifies \(P\), and thatif \(a\) exemplifies \(P\)then \(x\) and \(y\) stand in the relation \(R\). Finally, hereare some examples of quantified formulas:

The language we defined above issecond-order because the lastclause in the definition of the formulas sanctions both quantifiedformulas of the form \(\forall x\phi\) and of the form \(\forallF\phi\). In what follows, we employ the standard definitions of thefollowing formulas:

The first of the above defines the conjunction \(\phi\)and\(\psi\); the second defines the disjunction \(\phi\)or\(\psi\); the third defines the biconditional \(\phi\)if and onlyif \(\psi\) (which we often abbreviate asiff); and thelast defines the existentially quantified formulathere is an\(\alpha\)such that \(\phi\). It should be noted here thatinstead of using a linear string of symbols to express molecular andquantified formulas, Frege developed a two-dimensional notation forsuch formulas. Since we won’t be using Frege’s notationfor complex formulas in what follows, we need not spend timedescribing it here.^[1]

But even if we put aside Frege’s notation for complex formulas,it is important to point out that Frege didn’t use atomicformulas of the form \(Px\), \(Rxy\), etc., as we have done. Insteadof including \(n\)-place relation names and variables among hisprimitive terms, he included primitive function names and variables suchas \(f\), \(g\), \(h\), …, and used them tosignifyfunctions. That is, instead of distinguishing objectsandrelations, Frege distinguished objects fromfunctions. Though some developments of the modern predicatecalculus include function terms among the simple terms of thelanguage, we have not included them because we shall not need them inthe development of Frege’s Theorem.

It is also important to point out that Frege used functionalapplication‘\(f(x)\)’ to formcomplex names in his language and used these names to representnatural language statements. To see how, note that Frege would use theexpression ‘\(f(x)\)’ to denote the value of the function\(f\) for the argument \(x\). Since he also recognized two specialobjects he calledtruth-values (The True and The False), hedefined aconcept to be any function that always maps itsarguments to truth-values. For example, whereas‘\(x^{2}+3\)’ and ‘father-of\(x\)’ signify ordinary functions, the expressions ‘\(x\)is happy’ (which we might represent as ‘\(Hx\)’) and‘\(x \gt 5\)’ signify concepts. The former signifies aconcept which maps any object that is happy to The True and all otherobjects to The False; the latter signifies a concept that maps anyobject that is greater than 5 to The True and all other objects to TheFalse. In this way, ordinary language predications like ‘\(b\)is happy’ and ‘4 is greater than 5’, oncerepresented in Frege’s language as ‘\(Hb\)’ and‘\(4 \gt 5\)’, becomenames of truth-values.

For the purposes of understanding Frege’s Theorem, we can thinkof our 1-place relation terms as denoting, or ranging over, Fregeanconcepts. Once we do this, we can take the formula ‘\(Hb\)’ to mean that \(b\)falls under the conceptbeinghappy. But for the purposes of understanding Frege’sTheorem, it is not necessary to suppose, with Frege, that conceptslikebeing happy are functions from objects to truth values.So, in what follows, one should remember that whereas we can interpretthe atomic formula \(Fx\) to mean either that \(x\)exemplifiesthe 1-place relation (i.e., property) \(F\) or that \(x\) falls underthe concept \(F\), Frege would understand such formulas as instancesof functional application. Nevertheless, we’ll henceforth call1-place relationsconcepts. For all practical purposes then,we may use the symbols \(F\), \(G\), … as variables rangingover concepts and though we sometimes write ‘\(F(x)\)’instead of ‘\(Fx\)’ for perspicuity in parsing anexpression, we should still think of this as a predication.

Frege also supposed that when abinary function \(f\) (i.e., afunction of two arguments) always maps the arguments \(x\) and \(y\)to a truth value, \(f\) is arelation. So it should beremembered that when we use the expression‘\(Rxy\)’(or sometimes‘\(R(x,y)\)’) to assert that the objects \(x\) and \(y\)stand in the relation \(R\), Frege would say that \(R\) maps the pairof objects \(x\) and \(y\) (in that order) to The True. But again,this Fregean interpretation is not required for understandingFrege’s Theorem. In what follows, we shall sometimes write thesymbol that denotes a mathematical relation in the usual‘infix’ notation; for example, ‘\(\gt\)’ denotesthe greater-than relation in the expression ‘\(x \gty\)’.

Finally, it is important to mention that one can add the followingclause to the definition of theformulas of our second-orderlanguage so as to include formulas that express identity claims:

• If \(v_{1}\) and \(v_{2}\) are any object terms, \(v_{1} = v_{2}\)is a formula.

Thus, formulas such as ‘\(x = y\)’ are part of thesecond-order predicate calculus with identity. Frege, too, hadprimitive identity statements; for him, identity is a binary functionthat maps a pair of objects to The True whenever those objects are thesame object. So whereas we shall suppose that statements like‘\(2^{2} = 4\)’ are simply true assertions and statementslike ‘\(2^{2} = 3\)’ are simply false ones, Frege took‘\(2^{2} = 4\)’ to be a name of The True and took‘\(2^{2} = 3\)’ to be a name of The False. The statementform‘\(f(x) = y\)’ plays animportant role in Frege’s axioms and definitions, but we shallnot need to assert claims of this form in order to deriveFrege’s Theorem. Instead, we shall assume (a) that identity issimply a 2-place relation and (b) that a unary function \(f\) isreally a relation \(R\) that has the following property: \(Rxy \ampRxz \to y = z\) (i.e., that functions are relations that always relatetheir first argument to at most one second argument). We may call suchrelationsfunctional relations. In other words, when Fregeasserts \( f(x) = y\), we may represent this as asserting that \(f\)is a functional relation \(R\) such that \(Rxy\). This generalizes to\( n\)-place relations for \(n\geq2\). For example, where \(+\) is thebinary addition function of arithmetic, we may represent thearithmetic statement \(2+3 = 5\) in our language as a claim of theform \(+(2,3,5)\), where \(+\) is taken to be a 3-place functionalrelation that obeys the condition: \(+(x,y,z) \amp +(x,y,w) \to z =w\).

1.2 The Logic

The basic axioms and rules of inference governing statements in oursecond-order language are similar to those of the first-orderpredicate calculus with identity, though they’ve been extendedto apply to claims involving universal quantifiers binding relationvariables. Where \(\phi\), \(\psi\), and \(\chi\) are any formulas,\(\alpha\) any variable and \(\tau\) any term of the same type as\(\alpha\) (i.e., both are object terms or both are \(n\)-placerelation terms), then the following are the basic axioms and rules ofsecond-order logic:

• The axioms for propositional logic. E.g.,
  \(\phi \to (\psi \to \phi)\)
  \((\phi \to (\psi \to \chi)) \to ((\phi \to \psi) \to (\phi \to\chi))\)
  \((\neg\phi \to \neg\psi) \to ((\neg\phi \to \psi) \to \phi)\)
• Universal Instantiation: \(\forall\alpha\phi \to\phi^{\tau}_{\alpha}\), where both \(\phi^{\tau}_{\alpha}\) is theresult of uniformly substituting \(\tau\) for the free occurrences of\(\alpha\) in \(\phi\), and \(\tau\) is substitutable for \(\alpha\)(i.e., no variable free in \(\tau\) becomes bound by any quantifier in\(\phi^{\tau}_{\alpha}\)). E.g., where ‘a’ is an objectterm and ‘P’ is a 1-place relation term,
  \(\forall xPx \to Pa\)
  \(\forall FFa \to Pa\)
  (The corresponding principle, Existential Introduction, for theexistential quantifier, i.e., \(\phi^{\tau}_{\alpha}\to \exists \alpha\phi\), is derivable.)
• Quantifier Distribution:
  \(\forall\alpha(\phi \to \psi) \to (\phi \to \forall\alpha\psi),\)
  where \(\alpha\) is any variable that isn’t free in\(\phi\)
• Laws of Identity:
  \(x\eqclose x\)
  \(x\eqclose y \to (\phi \to \phi'),\)
  where \(\phi'\) is the result of substituting one or moreoccurrences of \(y\) for \(x\) in \(\phi\).
• Modus Ponens (MP): from \(\phi\) and \(\phi\to\psi\), we may infer\(\psi\).
• Rule of Generalization (GEN): from \(\phi\), we may infer\(\forall\alpha\phi\).

In what follows, we shall assume familiarity with the above axioms andrules as we derive Frege’s Theorem. As noted, these areessentially the same as the axioms for the first-order predicatecalculus, except for the addition of laws for the second-orderquantifiers \(\forall F\) and \(\exists F\) that correspond to thelaws governing the first-order quantifiers \(\forall x\) and \(\existsx\).

Some of the above laws are found explicitly inGg I, thoughexpressed in Frege’s notation. For example, inGgI, §47, we find Frege’s versions of thefollowing:

These are first introduced, however, inGg I,§§18, 20, 25, and 20, respectively.

Though Frege essentially had a second-order logic inGg, hisrules of inference don’t look as familiar, or as simple, as MPand GEN. The reason is that Frege’s rules of inference governnot only his graphical notation for molecular and quantified formulas,but also his special purpose symbols, such as certain lowercaseletters used as placeholders, certain Gothic letters and letters usedas bound variables, and various other signs of his system we have notyet mentioned. Since Frege’s notation for rules of inferencewill play no role in the discussion that follows, we shall againsimplify our task by not describing it further.

1.3 The Theory of Concepts

The modern second-order predicate calculus includes acomprehension principle that effectively guarantees that thereexists an \(n\)-place relation corresponding to every open formulawith \(n\) free object variables \(x_1,\ldots,x_n\). We introduce thisprinciple by considering the following 1-place case:

Comprehension Principle for Concepts:
\(\exists G \forall x(Gx \equiv \phi)\),
where \(\phi\) is anyformula in which \(G\) doesn’t occur free.

Similarly the following is a Comprehension Principle for 2-placeRelations:

Comprehension Principle for 2-place Relations:
\(\exists R\forall x\forall y(Rxy \equiv \phi)\),
where\(\phi\) is any formula in which \(R\) doesn’t occur free.

Although Frege didn’t explicitly formulate these comprehensionprinciples, they are derivable in his system and constitute veryimportant generalizations within his system that reveal its underlyingtheory of concepts and relations. We can see these principles at workby formulating the following instance of comprehension, where‘\(Ox\)’ asserts that \(x\) isodd:

\(\exists G\forall x(Gx \equiv (Ox \amp x \gt 5))\)

This asserts: there exists a concept \(G\) such that for every object\(x\), \(x\) falls under \(G\) if and only if \(x\) is odd and greaterthan 5. If our second-order language were extended to include theprimitive predicates ‘\(O\)’ and ‘\(\gt\)’ andthe primitive object term ‘5’, then the above instance ofthe Comprehension Principle for Concepts would be an axiom (and hence,theorem) of second-order logic.

Similarly, the following is an instance of the Comprehension Principlefor Relations:

\(\exists R\forall x\forall y(Rxy \equiv (Ox \amp x \gt y))\)

This asserts the existence of a relation that objects \(x\) and \( y\)bear to one another just in case the complex condition \(Ox \amp x \gty\) holds.

Logicians nowadays typically distinguish the open formula \(\phi\) inwhich the variable \(x\) is free from the corresponding name of aconcept. For example, they use the notation \([\lambda x \, Ox \amp x\gt 5]\) as the name of the complex conceptbeing an \(x\) suchthat \(x\) is odd and \(x\) is greater than 5 (or, morenaturally, ‘being odd and greater than 5’). Theterm-forming operator \(\lambda x\) (which we read as ‘being an\(x\) such that’) combines with a formula \(\phi\) in which\(x\) is free to produce \([\lambda x\,\phi]\). The\(\lambda\)-expression is a name of the concept expressed by theformula \(\phi\). In what follows, the scope of the variable-bindingoperator \(\lambda x\) in \([\lambda x\,\phi]\) applies to the entireformula \(\phi\), no matter how complex, so that instead of writing,for example, \([\lambda x\,(Ox \amp x \gt 5)]\), we shall simplywrite: \([\lambda x\, Ox \amp x \gt 5]\).

This notation can be extended for relations. The expression:

\([\lambda xy \, Ox \amp x \gt y]\)

names the 2-place relationbeing an \(x\)and \(y\)such that \(x\)is odd and \(x\)is greaterthan \(y\).

It is important to emphasize that Frege didn’t use\(\lambda\)-notation. By contrast, he thought that predicativeexpressions such as ‘(\(\, )\) is happy’ are incompleteexpressions and that the concepts they denoted wereunsaturated. We need not discuss Frege’s reasons forthis in this entry, though interested readers may consult his 1892essay “Concept and Object”.

For the purposes of understanding Frege’s Theorem, we only needto introduce one axiom that governs \(\lambda\)-notation, namely, theprinciple known as \(\lambda \)-Conversion. Let \(\phi\) be anyformula and let \(\phi^y_x\) be the result of substituting thevariable \(y\) for free occurrences of \(x\) everywhere in \(\phi\).Then the principle of \(\lambda\)-Conversion is:

\(\lambda\)-Conversion:
\(\forall y([\lambda x \, \phi]y \equiv \phi^y_x)\)

This asserts: for any object \(y\), \(y\) falls under the concept\([\lambda x \, \phi]\) if and only if \(y\) is such that\(\phi^y_x\). So, using our example, the following is an instance of\(\lambda\)-conversion:

\(\forall y([\lambda x \, Ox \amp x \gt 5]y \equiv Oy \amp y \gt 5)\)

This asserts: for any object \(y\), \(y\) falls under theconceptbeing odd and greater than 5 if and only if \(y\) isodd and greater than 5. Note that when the quantified variable \(y\)is instantiated to some object term, the resulting instance of\(\lambda\)-Conversion is a biconditional. Thus, among the manyconsequences of this axiom we find: 6 falls under theconceptbeing odd and greater than 5 if and only if 6 is oddand greater than 5 (in this case, the biconditional remains truebecause both sides are false).

Some logicians call the rule of inference derived from theright-to-left direction of such biconditionals‘\(\lambda\)-Abstraction’. For example, the inference fromthe premise:

\(O6 \amp 6 \gt 5\)

to the conclusion:

\([\lambda x \, Ox \amp x \gt 5]6\)

is justified by \(\lambda\)-Abstraction. (Here we have a case of avalid inference in which the premise and the conclusion are bothfalse.)

The principle of \(\lambda\)-Conversion can be generalized, so that itgoverns \(n\)-place \(\lambda\)-expressions as well. Here is the2-place case:

\(\forall z\forall w([\lambda xy\, \phi]zw \equiv \phi^{z,w}_{x,y})\)

(In this formula \(\phi^{z,w}_{x,y}\) is the result of simultaneouslysubstituting \(z\) for \(x\) and \(w\) for \(y\) in \(\phi\).)

The reader should construct an instance of this principle using ourexample \([\lambda xy \, Ox \amp x \gt y].\)

It should be noted at this point that instead of using comprehensionprinciples, Frege had a distinguished rule in his system that isequivalent to such principles, namely, his Rule of Substitution.Though Frege’s Rule of Substitution appears to allow thesubstitution of formulas \(\phi\) for free concept variables \(F\) intheorems of logic, we can understand this rule in terms of thesecond-order logic we’ve defined as follows: in any theorem oflogic with a free variable \(F^{n}\), one may both substitute any\(n\)-place \(\lambda\)-expression \([\lambda x_{1}\ldots x_{n}\,\phi]\) for \(F^{n}\) and then perform \(\lambda\)-conversion. Forexample, in the second-order system we now have, one can infer\(\forall x(Ox \amp x \gt 5 \equiv Ox \amp x \gt 5)\) from \(\forally(Fy \equiv Fy)\) by first substituting \([\lambda x \, Ox \amp x \gt5]\) for \(F\) and then using \(\lambda\)-Conversion on all theresulting subformulas containing the \(\lambda\)-expression that flankthe \(\equiv\) sign. Frege’s Rule of Substitution allows one todo all this in one step. Readers interested in learning a bit moreabout the connection between the Rule of Substitution andComprehension Principles described above can consult the followingsupplementary document:

Frege’s Rule of Substitution

Finally, it is important to point out that the system we have justdescribed, i.e., second-order logic with identity and comprehensionprinciples, extended with \(\lambda\)-expressions and\(\lambda\)-Conversion, is consistent. Its axioms are true even invery small interpretations, e.g., ones in which the domain of objectscontains a single object and each domain of \(n\)-place relations \((n\geq1)\) has just two relations. For example, if the domain ofobjects contains a single object, sayb, and the domain of1-place relations contains two concepts (i.e., one whichbfalls under and one which nothing falls under), then all of the aboveaxioms are true, including the Comprehension Principle for Conceptsand 1-place \(\lambda\)-Conversion. Even so, the system describedabove requires that every concept has a negation, every pair ofconcepts has a conjunction, every pair of concepts has a disjunction,etc. The reader should be able to write down instances of thecomprehension principle which demonstrate these claims.

Readers whose main goal is to understand Frege’s Theorem can nowskip directly to Section 3.

2. Frege’s Theory of Extensions: Basic Law V

Though the present section is not required for understanding the proofof Frege’s Theorem, we include it so that the reader can getsome sense of how second-order logic (with comprehension) gives riseto Russell’s paradox when one adds Frege’s theory ofcourses-of-values and extensions. Though we shall briefly discussFrege’s notation for courses-of-values, we’ll subsequentlyswitch to simpler notation for naming the extensions of concepts. Forthe purposes of this section, let us suppose that we have primitivefunction terms \(f\), \(g\), \(h\), … in our language and thatfunctional applications such as \(f(x)\), \(g(y)\), etc., areallowed.

The principle that undermined Frege’s system, Basic Law V, wasone that attempted to systematize the notions ‘course-of-valuesof a function’ and ‘extension of a concept’. Thecourse-of-values of a function \(f\) is something like a set ofordered pairs that records the value \(f(x)\) for every argument \(x\). For example, the course-of-values of the functionfather ofx records, among other things, that Bill Clinton is the value ofthe function when Chelsea Clinton is the argument. Thecourse-of-values for the function \(x^2\) records, among other things,that the number 4 is the value when the number 2 is the argument, that9 is the value when 3 is the argument, etc. When a function \(f\) is aconcept, Frege called the course-of-values for that concept itsextension. The extension of a concept is something like theset of all objects that fall under the concept, for the extensionrecords all of the objects that the concept maps to The True. Forexample, the extension of the concept \(x\)is a positive eveninteger less than 8 is something like the set consisting of thenumbers 2, 4, and 6.

2.1 Notation for Courses-of-Values of Functions

Frege introduces primitive notation for courses-of-values inGg I, §9. He switched to the lower case Greekletters \(\epsilon\) and \(\alpha\) when writing the names ofcourses-of-values and extensions, and placed smooth breathing marksover them to indicate they were variable-binding operators. So:

\(\stackrel{,}{\epsilon}\! f(\epsilon)\)

and

\(\stackrel{,}{\alpha}\! g(\alpha)\)

designate the course-of-values of the functions \(f\) and \(g\),respectively. In this notation, the symbols \(\stackrel{,}{\epsilon}\)and \(\stackrel{,}{\alpha}\) bind the object variables \(\epsilon\)and \(\alpha\) in the expressions \(f(\epsilon)\) and \(g(\alpha)\),respectively, and the resulting expression denotes acourse-of-values.

Here is a pair of examples of Frege’s notation forcourses-of-values. This pair of examples comes fromGgI, §9. Frege uses the expression:

\(\stackrel{,}{\epsilon}\! (\epsilon^{2}- \epsilon)\)

to denote the course-of-values of the function represented by the openformula:

\(x^{2} - x\)

He also uses:

\(\stackrel{,}{\alpha}\! (\alpha \cdot (\alpha - 1))\)

to denote the course-of-values of the function represented by the openformula:

\(x \cdot (x - 1)\)

Frege then notes that if the functions \(x^{2} - x\) and \(x \cdot (x- 1)\) map the same arguments to the same values, then the extensionsof those two functions are the same, and vice versa. That is, he notesthat:

\(\forall x[x^{2}-x = x \cdot (x - 1)]\)

holds if and only if:

\({\stackrel{,}{\epsilon}}(\epsilon^{2} - \epsilon) ={\stackrel{,}{\alpha}} (\alpha \cdot (\alpha - 1))\)

This equivalence will become embodied in Basic Law V. Indeed,Frege’s formulation of Basic Law V inGg I,§20 can now be represented in our language (temporarily extendedwith function terms and functional application) as follows:

Basic Law V:
\(\stackrel{,}{\epsilon}\! f(\epsilon) \eqclose \stackrel{,}{\alpha}\!g(\alpha) \equivwide \forall x[f(x) \eqclose g(x)]\)

This principle asserts: the course-of-values of the function \(f\) isidentical to the course-of-values of the function \(g\) if and only if\(f\) and g map every object to the same value. [Actually, Frege usesan identity sign instead of the biconditional sign as the mainconnective of the principle. The reason he could do this is that, inhis system, when two sentences are materially equivalent, theyname the same truth value.] We shall soon see why thisprinciple is inconsistent.

2.2 Notation for Extensions of Concepts

Since concepts, for Frege, are functions that always map theirarguments to a truth value, we may introduce some new notation to helpus represent Frege’s method of forming names of the extensionsof concepts. This new notation takes advantage of our\(\lambda\)-notation for naming concepts, and so allows us tointroduce a new kind of function term where Frege introduced avariable-binding operator.

Let us stipulate that where \(\Pi\) is any 1-place concept term (nameor variable), the notation ‘\(\epsilon\Pi\)’ designatesthe extension of the concept \(\Pi\). So, for example, \(\epsilon F\)denotes the extension of the concept \(F\). Note that 1-place\(\lambda\)-expressions of the form \([\lambda x\,\phi]\) are 1-placeconcept terms, and so \(\epsilon[\lambda x\,\phi]\) is well-formed anddesignates the extension of the concept \([\lambda x\,\phi]\). Thus,whereas Frege used \(\stackrel{,}{\epsilon}\) as a variable-bindingoperator that binds an object variable in a formula to produce thename of an extension, we are using \(\epsilon\) as a term-formingfunction symbol that applies to 1-place concept terms to produce termsdenoting, or ranging over, objects. Thus, when \(\epsilon\) isprefixed to a concept name, the resulting expression is a name of anobject, and in particular, a name of the extension of the conceptdenoted by the concept name. When the \(\epsilon\) is prefixed to aconcept variable, e.g., as in \(\epsilon F\), the resulting expressionis a kind of complex variable that ranges over extensions: for eachvalue of the variable \(F\), \(\epsilon F\) denotes the extension of\(F\).

Here is an example of our notation involving a pair of complexconcepts. Consider the conceptthat which when added to 4equals 5, or using \(\lambda\)-notation, the followingconcept:

\([\lambda x \, x+4 = 5]\)

We use the following notation to denote the extension of this concept:

\(\epsilon[\lambda x \, x+4 = 5]\)

Now consider the conceptthat which when added to \(2^{2}\)equals 5 (i.e., \([\lambda x~x+2^{2} = 5])\). We use thefollowing notation to denote the extension of this concept:

\(\epsilon[\lambda x \, x+2^{2} = 5]\)

Note that it seems natural to identify these two extensions given thatall and only the objects that fall under the first concept fall underthe second. Those readers already familiar with the\(\lambda\)-calculus should remember that \(\epsilon[\lambda x~\phi]\)denotes an object, that \([\lambda x~\phi]\) denotes a concept, andthat Frege rigorously distinguished objects and concepts and supposedthem to constitute mutually exclusive domains.

2.3 Membership in an Extension

If we remember that the extension of a concept is something like theset of objects that fall under the concept, then we could replaceFrege’s talk of ‘extensions’ by talk of‘sets’ and use the following ‘set notation’ torefer to the set of objects that when added to 4 yield 5 and the setof objects that when added to \(2^{2}\) yield 5, respectively:

Frege took advantage of his second-order language todefinewhat it is for an object to be a member of an extension or set.Although Frege used the notation \(x \cap y\) to designate themembership relation, we shall follow the more usual practice of using\(x\in y\). Thus, the following captures the main features ofFrege’s definition of membership inGg I,§34:^[2]

\(x\in y \eqdef \exists G (y\eqclose \epsilon G \amp Gx)\)

In other words, \(x\) is an element of \(y\) just in case \(x\) fallsunder a concept of which \(y\) is the extension. For example, giventhis definition, one can prove that John is a member of the extensionof the conceptbeing happy (formally: \(j \in \epsilon H)\)from the premise that John falls under the conceptbeinghappy (‘\(Hj\)’). Here is a simple proof:

Some readers may wish to examine a somewhat more complex example, inwhich the above definition of membership is used to prove that 1 \(\in\epsilon[\lambda x \, x+2^2 = 5]\) given the premise that \( 1+2^{2} =5\). (A More Complex Example)

Before we turn to Basic Law V, it is important to mention an importantfact about our representation of Frege’s system, in whichwe’ve introduced the term-forming operator \(\epsilon\) intosecond-order logic with identity. The resulting system has thefollowing principle, which asserts that every concept has anextension, as a theorem:

Existence of Extensions:
\(\forall G\exists x(x = \epsilon G)\)

To see that this is derivable given our work thus far, recall line 2of the proof in the above example: the laws of identity allow us toassert that:

\(\epsilon F = \epsilon F\)

In second-order logic with identity, this is an instance of \(x = x\)(strictly speaking, one first derives \(\forall x(x = x)\) from theaxiom \(x = x\) by GEN, and then instantiates the universallyquantified variable \(x\) to \(\epsilon F\)). So, by existentialgeneralization, it follows that:

\(\exists x(x = \epsilon F)\)

But now the Existence of Extensions principle follows by universalgeneralization on the concept variable \(F\). Thus, simply by adding aterm-forming operator such as \(\epsilon\) to classical logic withidentity, it is provable that every concept gets correlated with anextension. Basic Law V will not only imply, but also place a conditionon, this correlation.

2.4 Basic Law V for Concepts

We can now represent the special case of Frege’s Basic Law Vthat applies to concepts, using our \(\epsilon\) notation:

Basic Law V (Special Case):
\(\epsilon F\eqclose \epsilon G \equivwide \forall x(Fx \equiv Gx)\)

In this special case, Basic Law V asserts: the extension of theconcept \(F\) is identical to the extension of the concept \(G\) ifand only if all and only the objects that fall under \(F\) fall under\(G\) (i.e., if and only if the concepts \(F\) and \(G\) arematerially equivalent). In more modern guise, Frege’s Basic LawV asserts that the set of \(F\)s is identical to the set of \(G\)s ifand only if \(F\) and \(G\) are materially equivalent:

\(\{x\mid Fx\}\eqclose \{y\mid Gy\} \equivwide \forall z(Fz \equivGz)\)

The example discussed above can now be seen as an instance of BasicLaw V:

This asserts that the extension of the conceptthat which addedto 4yields 5 is identical to the extension of theconceptthat which added to \(2^{2}\)yields 5 ifand only if all and only the objects that when added to 4 yield 5 areobjects that when added to \(2^{2}\) yield 5.

There are two important corollaries to Law V that play a role in whatfollows: the Law of Extensions and the Principle of Extensionality.The Law of Extensions (cf.Gg I, §55, Theorem 1)asserts that an object is a member of the extension of a concept ifand only if it falls under that concept:

Law of Extensions:
\(\forall F \forall x(x \in\epsilon F \equiv Fx)\)

(Derivation of the Law of Extensions)

Basic Law V also correctly implies the Principle of Extensionality.This principle asserts that if two extensions have the same members,they are identical. Let us define ‘\(x\)is anextension’ as follows:

\(\mathit{Extension}(x) \eqdef \exists F (x = \epsilon F)\)

Then we may formally represent and derive the principle ofextensionality as follows:

Principle of Extensionality:
\(\mathit{Extension}(x) \amp \mathit{Extension}(y) \to [\forall z(z\in x\equiv z\in y) \to x\eqclose y]\)

(Derivation of the Principle of Extensionality)

The above facts about Basic Law V will be used in the next subsectionsto show why it maynot be consistently added to second-orderlogic with comprehension. Frege was made aware of the inconsistency byBertrand Russell, who sent him a letter formulating‘Russell’s Paradox’ just as the second volume ofGg was going to press. Frege quickly added anAppendix to the second volume, describing two distinct ways ofderiving a contradiction from Basic Law V. He also suggested a way ofrepairing Law V, but Quine (1995) later showed that such a repair wasdisastrous, since it would force the domain of objects to contain atmost one object.

In the next subsections, we describe the two ways of deriving acontradiction from Basic Law V that Frege described in the Appendix toGg. The first establishes the contradiction directly, withoutany special definitions. The second deploys the membership relationand more closely follows Russell’s Paradox. As we shall see, thefollowing combination is a volatile mix: (a) the ComprehensionPrinciple for concepts, which ensures that there is a conceptcorresponding to every formula with free variable \(x\), (b) theExistence of Extensions principle, which ensures every concept iscorrelated with an extension, and (c) Basic Law V, which ensures thatthe correlation of concepts with extensions behaves in a certainway.

2.5 First Derivation of the Contradiction

In the Appendix toGg II, Frege shows that acontradiction can be derived from Basic Law V once we formulate theconceptbeing an x that is the extension of some concept which xdoesn’t fall under. We may use the following\(\lambda\)-expression to represent this concept:

\([\lambda x \, \exists F (x\eqclose \epsilon F \amp \neg Fx)]\)

We know that there exists such a concept, since the open formula inthe scope of \(\lambda x\) can be used in the Comprehension Principlefor Concepts. Now by the Existence of Extensions principle, thefollowing concept exists and is correlated with it:

\(\epsilon[\lambda x \, \exists F(x\eqclose \epsilon F \amp \negFx)]\)

It can now be proved that this extension falls under the concept\([\lambda x \, \exists F(x\eqclose \epsilon F \amp \neg Fx)]\) if andonly if it does not.

(First Derivation of the Contradiction.)

2.6 Second Derivation of the Contradiction

In the Appendix toGg II, Frege also explains howBasic Law V implies the existence of the paradoxical Russell set. Wecan represent his reasoning as follows. From the Law of Extensions(which was derived from Basic Law V above), one can establish a NaiveComprehension Axiom for Extensions in three simple steps. First weinstantiate the Law of Extensions to the free variable \(F\), toyield:

\(\forall x(x \in \epsilon F\equiv Fx)\)

By existentially generalizing on \(\epsilon F\), it follows that:

\(\exists y\forall x(x \in y \equiv Fx)\)

Now at this point, we may universally generalize on the variable \(F\)to get the following second-order Naive Comprehension Axiom forextensions, which asserts that for every concept \(F\), there is anextension which has as members all and only the objects that fallunder \(F\):

Naive Comprehension Axiom for Extensions:
\(\forall F\exists y\forall x(x\in y \equiv Fx)\)

The Naive Comprehension Axiom gives rise to Russell’s Paradoxonce we instantiate the quantified variable \(F\) to the concept\([\lambda z \, z\notinclose z]\), where \(z \notinclose z\) simplyabbreviates \(\neg(z\in z)\), to yield:

\(\exists y\forall x(x\inclose y \equiv [\lambda z \, z\notinclosez]x)\)

By \(\lambda\)-Conversion, this is equivalent to:

\(\exists y\forall x(x\inclose y \equiv x \notinclose x)\)

(Note: Frege could have reached this last result in one step from\(\exists y\forall x(x\inclose y \equiv Fx)\) using his Rule ofSubstitution.)

The contradiction now goes as follows. Let \(b\) be such an objectasserted to exist by the claim we just derived. So we know:

\(\forall x(x\inclose b \equiv x\notinclose x)\)

But we can now instantiate the universally quantified variable to theobject \(b\) to yield the following contradiction:

\(b\inclose b \equiv b\notinclose b\)

(See the entry onRussell’s Paradox.)

2.7 How the Paradox is Engendered

We’ve now reconstructed the inconsistency in Frege’ssystem by representing his logic and Basic Law V in a modern system ofsecond-order logic. Philosophers have diagnosed the inconsistency invarious ways, and it is safe to say that the matter is still somewhatcontroversial. In this subsection, we discuss only the basic elementsof the problem. Most philosophers and logicians agree that the reasonsecond-order logic can’t be extended by Basic Law V is that theresulting system requires the impossible situation in which the domainof concepts has to be strictly larger than the domain of extensionswhile at the same time the domain of extensions has to be as large asthe domain of concepts.

To analyze the inconsistency in more detail, consideranextensional model of concepts, in which the materialequivalence of concepts \(F\) and \(G\) serves as both necessary andsufficient conditions for the identity of \(F\) and \(G\), i.e., inwhich \(F = G \equiv \forall x(Fx \equiv Gx)\). So, given thisunderstanding, if it isnot the case that \(F\) and \(G\) arematerially equivalent, then \(F\) and \(G\) are distinct concepts; andif \(F\) and \(G\) are distinct concepts, then they are not materiallyequivalent.

With this extensional view of concepts in mind, we can see how aparadox is engendered. Recall first that the Existence of Extensionsprinciple correlates each concept \(F\) with an extension \(\epsilonF\). Each direction of Basic Law V requires that this correlation havecertain properties. We shall see, for example, that the right-to-leftdirection of Basic Law V (i.e., Va) requires that no concept getscorrelated with two distinct extensions. [Frege uses the label‘Vb’ to designate the left-to-right direction of Basic LawV, and uses ‘Va’ for a variant of the right-to-leftdirection. See, for example,Gg I, §52. However,many commentators use ‘Va’ to designate the left-to-rightdirection. We shall follow Frege’s use, since that will makesense of his Appendix toGg II, in which he discussesthe paradoxes by discussing Vb and Va.] We may represent Frege’sVa as follows:

Basic Law Va:
\(\forall x(Fx \equiv Gx) \to\epsilon F = \epsilon G\)

So the contrapositive asserts that if \(\epsilon F \neq \epsilon G\),then \(\neg \forall x(Fx \equiv Gx)\). But in the case where thematerial equivalence of \(F\) and \(G\) is a necessary condition for\(F = G\), i.e., in the case where \(\neg \forall x(Fx \equiv Gx)\)implies \(F \neq G\), then Va implies that if \(\epsilon F \neq\epsilon G\), then \(F \neq G\), i.e., that whenever the extensions of\(F\) and \(G\) differ, the concepts with which they are correlated,namely \(F\) and \(G\), differ. This means that the correlationbetween concepts and extensions that Basic Law V sets up must be afunction – no concept gets correlated with two distinctextensions (though for all Va tells us, distinct concepts might getcorrelated with the same extension). Frege noted (in the AppendixtoGg II) that this direction of Basic Law Vdoesn’t seem problematic.

However, the left-to-right direction of Basic Law V (i.e., Vb) is moreserious. We may represent Vb as follows:

Basic Law Vb:
\(\epsilon F = \epsilon G \to\forall x(Fx \equiv Gx)\)

So the contrapositive asserts that if \(\neg \forall x(Fx \equiv Gx)\)then \(\epsilon F \neq \epsilon G\). But in the case where thematerial equivalence of \(F\) and \(G\) is a sufficient condition for\(F = G\), i.e., in the case where \(F \neq G\) implies \(\neg \forallx(Fx \equiv Gx)\), then Vb implies \(F \neq G \to \epsilon F \neq\epsilon G\), i.e., that if concepts \(F\) and \(G\) differ, theextensions of \(F\) and \(G\) differ. So, the correlation that BasicLaw V sets up between concepts and extensions will have to beone-to-one; i.e., it correlates distinct concepts with distinctextensions. Since every concept is correlated with some extension,there have to be at least as many extensions as there areconcepts.

But the problem is that second-order logic with Basic Law Vas awhole requires that there bemore concepts thanextensions. The requirement that there be more concepts thanextensions is imposed jointly by the Comprehension Principle forConceptsand the new significance this principle takes on inthe presence of Basic Law V. The Comprehension Principle for Conceptsasserts the existence of a concept for every condition on objectsexpressible in the language. Now although it may seem that thisprinciple, in and of itself, forces the domain of concepts to belarger than the domain of objects, it is a model-theoretic fact thatthere are models of second-order logic with the ComprehensionPrinciple for Concepts (but without Basic Law V) in which the domainof concepts is not larger than the domain of objects.^[3] However, the addition of Basic Law V to Frege’s system forcesthe domain of concepts to be larger than the domain of objects (and solarger than the domain of extensions), due to the endless cycle of newconcepts that arise in connection with the new extensions contributedby Basic Law V. However, as we saw in the last paragraph, Vb requiresthat there be at least as many extensions as there are concepts.

Thus, the addition of Basic Law V to second-order logic implies animpossible situation in which the domain of concepts has to bestrictly larger than the domain of extensions while at the same timethe domain of extensions has to be as large as the domain ofconcepts.

Recently, there has been a lot of interest in discovering ways ofrepairing the Fregean theory of extensions. The traditional view isthat one must either restrict Basic Law V or restrict theComprehension Principle for Concepts. Recently, Boolos (1986/87, 1993)developed one of the more interesting suggestions for revising BasicLaw V without abandoning second-order logic and its comprehensionprinciple for concepts. On the other hand, there have been manysuggestions for restricting the Comprehension Principle for Concepts.The most severe of these is to abandon second-order logic (and theComprehension Principle for Concepts) altogether. Schroeder-Heister(1987) conjectured that the first-order portion of Frege’ssystem (i.e., the system which results by adding Basic Law V to thefirst-order predicate calculus) was consistent and this was proved byT. Parsons (1987) and Burgess (1998).^[4] Heck (1996), Wehmeier (1999), Ferreira & Wehmeier (2002), andFerreira (2005) consider less drastic moves. They investigate systemsof second-order logic which have been extended by Basic Law V but inwhich the Comprehension Principle for Concepts is restricted in someway. See also Anderson & Zalta (2004) and Antonelli & May(2005) for different approaches to repairing Frege’s system. SeeFine (2002) for a discussion of the limits of Frege’s method andsee Burgess (2005) for a good general overview.

We will not discuss the above research further in the present entry,for none of these alternatives have achieved a clear consensus.Instead, we focus on the theoretical accomplishment revealed byFrege’s work inGg. As noted in theIntroduction, Frege validly proved a rather deep fact about thenatural numbers notwithstanding the inconsistency of Basic Law V. Hederived the Dedekind/Peano axioms for number theory in second-orderlogic from Hume’s Principle (which was briefly mentioned aboveand which will be discussed in the next section). But this fact wentunnoticed for many years. Though Geach (1955) claimed such aderivation was possible, C. Parsons (1965) was the first to note thatHume’s Principle was powerful enough for the derivation of theDedekind/Peano axioms. Though Wright (1983) actually carried out mostof the derivation, Heck (1993) showed that although Frege did useBasic Law V to derive Hume’s principle, his (Frege’s)subsequent derivations of the Dedekind/Peano axioms of number theoryfrom Hume’s Principle never made anessential appeal toBasic Law V. Since Hume’s Principle can be consistently added tosecond-order logic, we may conclude that Frege himself validly derivedthe basic laws of number theory. It will be the task of the next fewsections to explain Frege’s accomplishments in this regard. Wewill do this in two stages. In §3 we study Frege’s attemptto derive Hume’s Principle from Basic Law V by analyzingcardinal numbers as extensions. Then, we put this aside in §4 and§5 to examine how Frege was able to derive the Dedekind/Peanoaxioms of number theory from Hume’s Principle alone.

3. Frege’s Analysis of Cardinal Numbers

Cardinal numbers are the numbers that can be used to answer thequestion ‘How many \(\ldots\) are there?’, and Fregediscovered that such numbers bear an interesting relationship to thenatural numbers. Frege’s insights concerning this relationshiptrace back to his work inGl, in which the notion ofan extension played very little role. The seminal idea ofGl §46 was the observation that a statement ofnumber (e.g., “There are eight planets”) is an assertionabout a concept. To explain this idea, Frege noted that one and thesame external phenomenon can be counted in different ways; forexample, a certain external phenomenon could be counted as 1 army, 5divisions, 25 regiments, 200 companies, 600 platoons, or 24,000people. Each way of counting the external phenomenon corresponds to amanner of its conception. The question “How many arethere?” is only properly formulated as the question “Howmany \(F\)s are there?” where a concept \(F\) is supplied. OnFrege’s view, the statements of number which answer suchquestions (e.g., “There are \(n\) \(F\)s”) tell ussomething about the concept involved. For example, the statement“There are eight planets in the solar system” tells usthat the ordinary conceptplanet in the solar system fallsunder thesecond-level numerical conceptbeingexemplified by eight objects.

InGl, Frege then moves from this realization, in whichstatements of numbers are analyzed as predicating second-levelnumerical concepts of first-level concepts, to develop an account ofthe cardinal and natural numbers as ‘self-subsistent’objects. He introduces a ‘cardinality operator’ onconcepts, namely, ‘the number belonging to the concept \(F\)’, which designates the cardinal number which numbers theobjects falling under \(F\). In what follows, we say this more simplyas ‘the number of \(F\)s’ and use the simple notation‘\(\#F\)’. Note that the operator # behaves like the\(\epsilon\) operator – when it is prefixed to a concept namelikeplanet (\(P\)), then \(\#P\) (“the number ofplanets”) denotes an object; when it is prefixed to a variablelike \(F\), then \(\# F\) ranges over the domain of objects (for eachconcept that \(F\) can take as a value, \(\#F\) denotes an objectrelative to that concept). Frege offers both an implicit (i.e.,contextual) and an explicit definition of this operator inGl. Both of these definitions require a preliminarydefinition of when two concepts \(F\) and \(G\) are in one-to-onecorrespondence or ‘equinumerous’. The notion ofequinumerosity plays an important and fundamental role in thedevelopment of Frege’s Theorem. After developing the definitionof equinumerosity, we then discuss Frege’s implicit and explicitdefinition of the number of \(F\)s. Only the former is needed for theproof of Frege’s Theorem, however.

3.1 Equinumerosity

In order to state the definition of equinumerosity, we shall employthe well-known logical notion ‘there exists a unique \(x\) suchthat \(\phi\)’. To say that there exists a unique \(x\) suchthat \(\phi\) is to say: there is some \(x\) such that \(\phi\), andanything \(y\) which is such that \(\phi\) is identical to \(x\). Inwhat follows, we use the notation ‘\(\exists!x\phi\)’ toabbreviate this notion of a formula being uniquely satisfied, and wedefine it formally as follows (where again, \(\phi^y_x\) is the resultof substituting \(y\) for the free occurrences of \(x\) in \(\phi\):

\(\exists!x\phi \eqdef \exists x[\phi \amp\forall y(\phi^y_x \to y =x)]\)

Now, in terms of this logical notion of unique existence, we can statea definition of equinumerosity that is weaker than the one Frege givesinGl (§§71, 72) but which nevertheless does thejob:^[5]

\(F\) and \(G\) areequinumerous just in case there is arelation \(R\) such that: (1) every object falling under \(F\) is\(R\)-related to a unique object falling under \(G\), and (2) everyobject falling under \(G\) is such that there is a unique objectfalling under \(F\) which is \(R\)-related to it.

In other words, \(F\) and \(G\) are equinumerous just in case there isa relation that establishes a one-to-one correspondence between the\(F\)s and the \(G\)s. If we let ‘\(F \approx G\)’ standfor equinumerosity, then the definition of this notion can be renderedformally as follows:

To see that Frege’s definition of equinumerosity workscorrectly, consider the following two examples. In the first example,we have two concepts, \(F\) and \(G\), that are equinumerous:

Figure 1

Although there are several different relations \(R\) which demonstratethe equinumerosity of \(F\) and \(G\), the particular relation used inFigure 1 is:

\(R_{1} = [\lambda xy \, (x\eqclose a\amp y\eqclose f) \lor (x\eqcloseb \amp y\eqclose g) \lor (x\eqclose c\amp y\eqclose e)]\)

It is a simple exercise to show that \(R_{1}\), as defined, is a‘witness’ to the equinumerosity of \(F\) and \(G\)(according to the definition).

In Figure 2, we have two concepts that are not equinumerous:

Figure 2

In this example, no relation \(R\) can satisfy the definition ofequinumerosity.

Given the discussion so far, it seems reasonable to suggest thatconcepts \(F\) and \(G\) will be equinumerous whenever the number ofobjects falling under \(F\) is identical to the number of objectsfalling under \(G\). This suggestion will be codified by Hume’sPrinciple. However, before discussing this principle, the readershould convince him- or herself of the following four facts: (1) thatthe material equivalence of two concepts implies their equinumerosity,(2) that equinumerosity is reflexive, (3) that equinumerosity issymmetric, and (4) that equinumerosity is transitive. In formal terms,the following facts are provable:

Facts About Equinumerosity:
1. \(\forall x(Fx \equiv Gx) \to F\apprxclose G \)
2. \(F \approx F\)
3. \(F\apprxclose G \to G\apprxclose F\)
4. \(F\apprxclose G \amp G\apprxclose H \to F\apprxclose H\)

The proofs of these facts, in each case, require the identification ofa relation that is a witness to the relevant equinumerosity claim. Insome cases, it is easy to identify the relation in question. In othercases, the reader should be able to ‘construct’ suchrelations (using \(\lambda\)-notation) by considering the examplesdescribed above. Facts (2) – (4) establish that equinumerosityis an ‘equivalence relation’ which divides up the domainof concepts into ‘equivalence classes’ of equinumerousconcepts.

3.2 Contextual Definition of ‘The Number of \(F\)s’: Hume’s Principle

InGl, Frege contextually defined ‘the number of\(F\)s’ in terms of the principle now known as Hume’s Principle:^[6]

Hume’s Principle:
The number of \(F\)s is identical to the number of \(G\)s if and onlyif \(F\) and \(G\) are equinumerous.

Using our notation ‘\(\#F\)’ to abbreviate ‘thenumber of \(F\)s’, we may formalize Hume’s Principle asfollows:

Hume’s Principle:
\(\#F\eqclose \#G \equiv F \approx G\)

Hume’s Principle is taken to be a contextual definition of\(\#F\) when the latter is assumed as a primitive notion governed bythe principle: the principle doesn’t explicitly define‘\(\#F\)’, but contextually defines it by definingcontexts (in this case, identity statements) in which it occurs.^[7] As we shall see, Hume’s Principle is the basic principle upon whichFrege forged his development of the theory of natural numbers. InGl, Frege sketched the derivations of the basic lawsof number theory from Hume’s Principle; these sketches weredeveloped into more rigorous proofs inGg I. We willexamine these derivations in the following sections.

Once Frege had a contextual definition of \(\#F\), he then defined acardinal number as any object which is the number of some concept:

\(x \textit{ is a cardinal number} \eqdef \exists F(x = \#F)\)

This represents the definition that appears inGl,§72.

Notice that Hume’s Principle bears an obvious formal resemblanceto Basic Law V. Both are biconditionals asserting the equivalence ofan identity among singular terms (the left-side condition) with anequivalence relation on concepts (the right-side condition). Indeed,both correlate concepts with certain objects. In the case ofHume’s Principle, each concept \(F\) is correlated with \(\#F\).However, whereas Basic Law V problematically requires that thecorrelation between concepts and extensions be one-to-one,Hume’s Principle only requires that the correlation betweenconcepts and numbers be many-to-one. Hume’s Principle oftencorrelates distinct concepts with the same number. For example, thedistinct conceptsauthor of Principia Mathematica(‘\([\lambda x \, Axp]\)’)andpositive integer between 1and 4 (‘\([\lambda x \, 1\lt x\lt 4]\)’) areequinumerous (both have two objects falling under them). So\(\#[\lambda x\, Axp]\) = \(\#[\lambda x \, 1\lt x\lt 4]\). Thus,Hume’s Principle, unlike Basic Law V, does not require that thedomain of numbers be as large as the domain of concepts. Indeed,several authors have developed models that show Hume’s Principlecan be consistently added to second-order logic. See the independentwork of Geach (1976, 446–7), Hodes (1984, 138), Burgess (1984)and Hazen (1985).

3.3 Explicit Definition of ‘The Number of \(F\)s’

[Note: The remaining two subsections are not strictly necessary forunderstanding the proof of Frege’s Theorem. They are includedhere for those who wish to have a more complete understanding of whatFrege in fact attempted to do. They presuppose the material in§2. Readers interested in just the positive aspects ofFrege’s accomplishments should skip directly to §4.]

Before we examine the powerful consequences that Frege derived fromHume’s Principle, it is worth digressing to describe his attemptto define ‘\(\#F\)’ explicitly and to derive Hume’sPrinciple from Basic Law V. The idea behind this attempt was therealization that if given any concept \(F\), the notion ofequinumerosity can be used to define the second-level conceptbeing a concept G that is equinumerous to F. Frege found away to collect all of the concepts equinumerous to a given concept\(F\) into a single extension. InGl §68, heinformally took this to be an extension consisting of first-orderconcepts by stipulating that the number of \(F\)s is the extension ofthe second-level concept:being a first-level concept equinumerousto F.

In terms of the example used at the end of the previous subsection,this informal definition identifies the number of the conceptauthor of Principia Mathematica as the extension consistingof all and only those first-level concepts that are equinumerous tothis concept; this extension has both \([\lambda x \, Axp]\) and\([\lambda x \, 1\lt x\lt 4]\) as members. Frege in fact identifiesthe cardinal number 2 with this extension, for it contains all andonly those concepts under which two objects fall. Similarly, Fregeidentifies the cardinal number 0 with the extension consisting of allthose first-level concepts under which no object falls; this extensionwould include such concepts asunicorn,centaur,prime number between 3and 5, etc. Frege’sinsight here inspired Russell to develop a somewhat similar definitionin his work, and it is now common to see references to the so-called“Frege-Russell definition of the cardinal numbers” asclasses of equinumerous concepts or sets.^[8] Of course, this explicit definition of ‘the number of \(F\)s’ stands or falls with a coherent conception of‘extension’. We know that Basic Law V does not offer sucha coherent conception.

3.4 Derivation of Hume’s Principle

Frege’s derivations of Hume’s Principle were invalidatedby the fact that it appeals to the inconsistent Basic Law V.Neverthelss, we briefly describe in this subsection, for interestedreaders, Frege’s derivations. InGl, §73,Frege sketches an informal proof of the right-to-left direction ofHume’s Principle using the above informal definition of thenumber of \(F\)s. The derivation appeals to the fact that a concept\(G\) is a member of the extension of the second-level conceptconcept equinumerous to F if and only if \(G\) isequinumerous to \(F\). In other words, the proof relies on a kind ofhigher-order version of the Law of Extensions (described above), theordinary version of which we know to be a consequence of Basic Law V.^[9] Here is a reconstruction of Frege’s proof inGl, §73, extended so as to cover both directionsof Hume’s Principle.

Reconstruction of theGrundlagen Derivation of Hume’s Principle

However, in the development ofGg, Frege didn’tformulate the extensions of second-level concepts. InGg,extensions donot contain concepts as members but ratherobjects. So Frege had to find another way to express the explicitdefinition described in the previous subsection. His technique was tolet extensions go proxy for their corresponding concepts. Since a fullreconstruction of this technique and the proof of Hume’sPrinciple inGg would constitute a digression for thepresent exposition, we shall describe the details for interestedreaders in a separate document:

Reconstruction of theGrundgesetze ‘Derivation’ of Hume’s Principle

Interestingly, Tennant (2004) and May & Wehmeier (2019)point out that inGg, Frege does not, in actual fact, deriveHume’s Principle as a biconditional. Instead, he derives bothdirections separately without combining them or indicating that thetwo directions should be conceived as a biconditional. Finally, asnoted on several occasions, the inconsistency in Basic Law Vinvalidated Frege’s derivation of Hume’s Principle. ButHume’s Principle, in and of itself, is a powerful and consistentprinciple.

4. Frege’s Analysis of Predecessor, Ancestrals, and the Natural Numbers

In what follows, we shall suppose that the second-order predicatecalculus with which we began has been extended with (a) a primitive\(\#\) operator, so that we can formulate terms such as \(\#F\) tosignify ‘the number of \(F\)s’, and (b) a new axiom,namely, Hume’s Principle, to govern the new terms. As previouslymentioned, Frege’s Theorem is that the Dedekind/Peano axioms ofnumber theory are derivable as theorems in a second-order predicatecalculus extended in this way. In this section, we introduce thedefinitions required for the proof of Frege’s Theorem. In thenext section, we go through the proof. In the final section, weconclude with a discussion of the philosophical questions that arisewhen we extend the predicate calculus in this way, and takeHume’s Principle as a replacement for Basic Law V.

Before we turn to the definitions required for the proof ofFrege’s Theorem, it would serve well to discuss one other groupof insights underlying Frege’s analysis of numbers. The first isthat the following series of concepts has a rather interestingproperty:

The interesting property of this series is that for each concept \(C_k\), all and only thenumbers of the concepts preceding\(C_k\) in the sequence fall under \(C_k\). So, for example, theconcepts preceding \(C_3\) are \(C_0\), \(C_1\), and \( C_2\).Accordingly, all and only the following numbers fall under \(C_3:\#C_0, \#C_1\), and \(\#C_2\).

Frege’ next insight was that these concepts can be used,respectively, to define the finite cardinal numbers, as follows:

This insight, however, led to another. Frege realized that though wemay identify this sequence of numbers with the natural numbers, such asequence is simply a list: it does not constitute a definition of aconcept (e.g.,natural number) that applies to all and onlythe numbers defined in the sequence. Such a concept is required if weare to proveas theorems the following axioms ofDedekind/Peano number theory:

Moreover, Frege recognized the need to employ the Principle ofMathematical Induction in the proof that every number has a successor.One cannot prove the claim thatevery number has a successorsimply by producing the sequence of expressions for cardinal numbers(e.g., the second of the two sequences described above). All such asequence demonstrates is that for every expression listed in thesequence, one can define an expression of the appropriate form tofollow it in the sequence. This is not the same as proving thatevery natural number has a successor.

4.1 Predecessor

To accomplish these further goals, Frege proceeded(Gl, §76, andGg I, §43)by defining the concept \(x\) (immediately)precedes\(y\):

\(x\) (immediately)precedes \(y\) if and only ifthere is a concept \(F\) and an object \(w\) such that: (a) \(w\)falls under \(F\), (b) \(y\) is the number of \(F\)s, and (c) \( x\)is the number of the conceptobject falling under \(F\) other thanw.

We may represent Frege’s definition formally in our language asfollows:

To illustrate this definition, let us temporarily assume that we knowsome facts about the natural numbers 1 and 2 to show that thedefinition properly predicts that \(\mathit{Precedes}(1,2)\), eventhough we haven’t yet defined these natural numbers. Let theexpression‘\([\lambda z \,Azp]\)’ denote the conceptauthor of Principia Mathematica. Only Bertrand Russell(‘\(r\)’) and Alfred Whitehead fall under this concept.Let the expression‘\([\lambda z \, Azp \ampz\neqclose r]\)’ denote the conceptauthor ofPrincipia Mathematica other thanRussell.^[10]Then the following may, for the purposes of this example, be taken asfacts:

• Russell falls under the conceptauthor of PrincipiaMathematica, i.e.,
      \([\lambda z \, Azp]r\)
• 2 is the number of the conceptauthor of PrincipiaMathematica, i.e.,
      \(2 = \#[\lambda z \, Azp]\)
• 1 is the number of the conceptauthor of Principia Mathematicaother than Russell, i.e.,
      \(1 = \#[\lambda z \, Azp \amp z\neqclose r] \)

If we assemble these truths into a conjunction and apply existentialgeneralization in the appropriate places, the result is the definiensof the definition of predecessor instantiated to the numbers 1 and 2.Thus, if given certain facts about the number of objects falling underthe certain concepts, the definition of predecessor correctly predictsthat \(\mathit{Precedes}(1,2)\).

4.2 The Ancestral of a RelationR

Frege next defines the relationx is an ancestor of y in theR-series. This new relation is called ‘the ancestral of therelationR’ and we henceforth designate this relationas \(R^*\). Frege first defined the ancestral of a relationRinBegr (Part III, Proposition 76), though the term‘ancestral relation’ comes to us from Whitehead andRussell 1910–1913 (I, *90·01). Frege’s phrase forthe ancestral is: “\(x\) comes before \(y\) in the\(R\)-series”; alternatively, “\(y\) follows \(x\) in the\(R\)-series”. (See alsoGl, §79, andGg I, §45.) The intuitive idea is easily graspedif we consider the relation \(x\)is the father of\(y\). Suppose that \(a\) is the father of \(b\), that \(b\) is thefather of \(c\), and that \(c\) is the father of \(d\). Then‘\(x\) is an ancestor of \(y\) in the fatherhood-series’is defined so that \(a\) is an ancestor of \(b\), \(c\), and \(d\),that \(b\) is an ancestor of \(c\) and \(d\), and that \(c\) is anancestor of \(d\).

Frege’s definition of the ancestral of R requires a preliminarydefinition:

F is hereditary in the R-series if and only if every pair of\(R\)-related objects \(x\) and \(y\) are such that \(y\) falls under\(F\) whenever \(x\) falls under \(F\)

In formal terms:

\(\mathit{Her}(F,R) \eqabbr \forall x\forall y(Rxy \to (Fx \to Fy))\)

Intuitively, the idea is that \(F\) is hereditary in theR-series if \(F\) is always ‘passed along’ from\(x\) to \(y\) whenever \(x\) and \(y\) are a pair ofR-related objects. (We warn the reader here that the notation‘\(\mathit{Her}(F,R)\)’ is merely an abbreviation of amuch longer statement. It isnot a formula of our languagehaving the form ‘\(R(x,y)\)’. In what follows, wesometimes introduce other such abbreviations.)

Frege’s definition of the ancestral of \(R\) can now be statedas follows:

x comes before y in the R-series \(\eqdef\)y fallsunder all thoseR-hereditary conceptsF under whichfalls every object to whichx isR-related.

In other words, \(y\) follows \(x\) in the R-series whenever \(y\)falls under everyR-hereditary concept \(F\) that isexemplified by everything immediatelyR-related to \(x\). Informal terms:

\(R^*(x,y) \eqdef \forall F[(\forall z(Rxz \to Fz) \amp\mathit{Her}(F,R)) \to Fy]\)

For example, Clinton’s father stands in the relationfather*of (i.e.,forefather) to Chelsea because she falls underevery hereditary concept that Clinton and his brother inherited fromClinton’s father. However, Clinton’s brother is not one ofChelsea’s forefathers, since he fails to be her father, hergrandfather, or any of the other links in the chain of fathers fromwhich Chelsea descended.

It is important to grasp the differences between a relation \(R\) andits ancestral \(R^*\).Rxy implies \(R^*(x,y)\) (e.g., ifClinton is a father of Chelsea, then Clinton is a forefather ofChelsea), but the converse doesn’t hold (Clinton’s fatheris a father* of Chelsea, but he is not a father of Chelsea). Indeed, agrasp of the definition of \(R^*\) should leave one able to prove thefollowing easy consequences, many of which correspond to theorems inBegr andGg:^[11]

Facts About \(R^*\):

1. \(Rxy \to R^*(x,y)\)
2. \(\neg\forall R\forall x\forall y(R^*(x,y)\to Rxy)\)
3. \([R^*(x,y) \amp \forall z(Rxz \to Fz) \amp \mathit{Her}(F,R)] \to Fy\)^[12]
4. \(R^*(x,y) \to \exists z \, Rzy\)
5. \([Fx \amp R^*(x,y) \amp \mathit{Her}(F,R)] \to Fy\)
6. \(Rxy \amp R^*(y,z) \to R^*(x,z)\)
7. \(R^*(x,y) \amp R^*(y,z) \to R^*(x,z)\)

The reader should consider what happens when \(R\) is taken to be therelation (immediately)precedes. Appealing to ourintuitive grasp of the numbers, we can say that it is an instance ofFact (1) that if 10 precedes 11, then 10 precedes* 11. Moreover,precedes is a witness to Fact 2: that 10 precedes* 12 does not implythat 10 precedes 12. The transitivity of precedes* is an instance ofFact (7). Below, when we restrict ourselves to the natural numbers, itbecomes intuitive to think of the difference between precedes andprecedes* as the difference betweenimmediately precedes andless-than.

4.3 The Weak Ancestral ofR

Given the notion of the ancestral of relation \(R\), Frege thendefines its weak ancestral, which he termed “\(y\) is a memberof the \(R\)-series beginning with \(x\)” (cf.Begr, Part III, Proposition 99;Gl,§81, andGg I, §46):

\(y\) is a member of the \(R\)-series beginning with \(x\) ifand only if either \(x\) bears the ancestral of \(R\) to \(y\) or \(x= y\).

In formal terms:

\(R^{+}(x,y) \eqdef R^*(x,y) \lor x\eqclose y\)

Frege would also read \(R^{+}(x,y)\) as: \(x\) is a member of theR-series ending with \(y\). Logicians call \(R^{+}\) the‘weak-ancestral’ of \(R\) because it is a weakened versionof \(R^*\). When we define the natural numbers below, and take \(R\)to beprecedes, we can intuitively regard its weak ancestral,precedes\(^{+}\), as the relationless-than-or-equal-to on the natural numbers.

The general definition of the weak ancestral of \(R\) yields thefollowing facts, many of which correspond to theorems inGg:^[13]

Facts About \(R^{+}\):

0. \(R^*(x,y) \to R^{+}(x,y)\)
1. \(Rxy \to R^{+}(x,y)\)
2. \(Rxy \amp R^{+}(y,z) \to R^*(x,z)\)
3. \(R^{+}(x,y) \amp Ryz \to R^*(x,z)\)
4. \(R^*(x,y) \amp Ryz \to R^{+}(x,z)\)
5. \(R^{+}(x,x)\)       (Reflexivity)
6. \(R^*(x,y) \to \exists z[R^{+}(x,z) \amp Rzy]\)
   (Proof of Fact 6 Concerning the Weak Ancestral)
7. \([Fx \amp R^{+}(x,y) \amp \mathit{Her}(F,R)] \to Fy\)
8. \(R^*(x,y) \amp Rzy \amp R \text{ is 1-1} \to R^{+}(x,z)\)^[14]

The proofs of these facts are left as exercises.

4.4 The ConceptNatural Number

Frege’s definition ofnatural number requires one morepreliminary definition. Frege identified the number 0 as the number ofthe conceptbeing non-self-identical. That is:

\(0 \eqdef \#[\lambda x \, x\neq x]\)

Since the logic of identity guarantees that no object isnon-self-identical, nothing falls under the conceptbeingnon-self-identical. Had Frege’s explicit definition of the\(\#F\) worked as he had intended, the number 0 would, in effect, beidentified with the extension consisting of all those extensions ofconcepts under which nothing falls. However, for the present purposes,we may note that 0 is defined in terms of (a) the primitive notion‘the number of \(F\)s’ and (b) a concept \(([\lambda x \,x\neq x])\) whose existence is guaranteed by our second-order logicwith identity and comprehension. It is straightforward to prove thefollowing Lemma Concerning Zero from this definition of 0:

Lemma Concerning Zero:
\(\#F\eqclose 0 \equivwide \neg\exists xFx\)

(Proof of Lemma Concerning Zero)

Note that the proof appeals to Hume’s Principle and facts aboutequinumerosity.

Frege’s definition of the conceptnatural number cannow be stated in terms of the weak-ancestral of Predecessor:

x is a natural number if and only if \(x\) is a member of thepredecessor-series beginning with 0

This definition appears inGl, §83, andGg I, §46 as the definition of ‘finitenumber’. Indeed, the natural numbers are precisely the finitecardinals. In formal terms, Frege’s definition becomes:

\(Nx ~ \eqdef ~ \mathit{Precedes}^{+}(0,x)\)

In what follows, we shall sometimes use the variables \(m\), \(n\),and \(o\) to range over the natural numbers. In other words,we’ll use formulas of the form \(\forall n(\ldots n\ldots)\) toabbreviate formulas of the form \(\forall x(Nx \to \ldots x\ldots)\),and use formulas of the form \(\exists n(\ldots n\ldots)\) toabbreviate formulas of the form \(\exists x(Nx \amp \ldotsx\ldots)\).

5. Frege’s Theorem

Frege’s Theorem is that the five Dedekind/Peano axioms fornumber theory can be derived from Hume’s Principle insecond-order logic. In this section, we reconstruct the proof of thistheorem; it can be extracted from Frege’s work using thedefinitions and theorems assembled so far. Some of the steps in thisproof can be found inGl. (See the Appendix to Boolos1990 for a reconstruction.) Our reconstruction follows Frege’sGg in spirit and in most details, but we have triedto simplify the presentation in several places. For a stricterdescription of Frege’sGg proof, the reader isreferred to Heck 1993. The following should help prepare the readerfor Heck’s excellent essay.

5.1 Zero is a Natural Number

The statement that zero is a natural number is an immediateconsequence of the definition ofnatural number:

Theorem 1:
\(N0\)

Proof: It is a simple consequence of the definition of‘weak ancestral’ that \(R^{+}\) is reflexive (see Fact 4about \(R^{+}\) in our subsection on the Weak Ancestral in §4).So \(\mathit{Precedes}^{+}(0,0)\). Hence, by the definition of naturalnumber, \(0\) is a natural number.

It seems that Frege never actually identified this fact explicitly inGl or labeled this fact as a numbered Theorem inGg I.

5.2 Zero Isn’t the Successor of Any Natural Number

It is also a simple consequence of the foregoing that 0 doesn’tsucceed any natural number. This can be represented formally asfollows:

Theorem 2:
\(\neg\exists x(Nx \amp \mathit{Precedes}(x,0))\)

Proof: Assume, forreductio, that some number, say\(n\), is such that \(\mathit{Precedes}(n,0)\). Then, by thedefinition of predecessor, it follows that there is a concept, say\(Q\) and an object, say \(c\), such that \(Qc \amp 0\eqclose \#Q \ampn\eqclose \#[\lambda z\, Qz \amp z\neq c]\). But by the LemmaConcerning Zero (above), \(0 = \#Q\) implies \(\neg\exists xQx\),which contradicts the fact that \(Qc\).

SeeGl, §78, Item (6); andGgI, §109, Theorem 126.

5.3 No Two Natural Numbers Have the Same Successor

The fact that no two natural numbers have the same successor issomewhat more difficult to prove (cf.Gl, §78,Item (5);Gg I, §95, Theorem 89). We mayformulate this theorem as follows, with \(m\), \(n\), and \(o\) asrestricted variables ranging over the natural numbers:

Theorem 3:
\(\forall m\forall n\forall o[\mathit{Precedes}(m,o)\amp\mathit{Precedes}(n,o) \to m = n]\)

In other words, this theorem asserts that predecessor is a one-to-onerelation on the natural numbers. To prove this theorem, it suffices toprove that predecessor is a one-to-one relation full stop. One canprove that predecessor is one-to-one from Hume’s Principle, withthe help of the following Equinumerosity Lemma, the proof of which israther long and involved. The Equinumerosity Lemma asserts that when\(F\) and \(G\) are equinumerous, \(x\) falls under \(F\), and \( y\)falls under \(G\), then the conceptobject falling under F otherthan x is equinumerous to the conceptobject falling under Gother than \(y\). The picture is something like this:

Figure 3

In terms of Figure 3, the Equinumerosity Lemma tells us that if thereis a relation \(R\) which is a witness to the equinumerosity of \( F\)and \(G\), then there is a relation \(R'\) which is a witness to theequinumerosity of the concepts that result when you restrict \( F\)and \(G\) to the objects other than \(x\) and \(y\), respectively.

To help us formalize the Equinumerosity Lemma, let \(F^{-x}\)abbreviate the concept \([\lambda z \, Fz \amp z\neq x]\) and let\(G^{-y}\) abbreviate the concept \( [\lambda z \, Gz \amp z \neqy]\). Then we have:

Equinumerosity Lemma:
\(F\apprxclose G \amp Fx \amp Gy \to F^{-x}\apprxclose G^{-y}\)

(Proof of Equinumerosity Lemma)

Now we can prove that Predecessor is a one-to-one relation from thisLemma and Hume’s Principle (cf.Gg I,§108):

So, if Predecessor is a one-to-one relation, it is a one-to-onerelation on the natural numbers. Therefore, no two numbers have thesame successor. This completes the proof of Theorem 3.

It is important to mention here that not only is Predecessor aone-to-one relation, it is also a functional relation:

Predecessor is a Functional Relation:
\(\forall x\forall y\forall z[\mathit{Precedes}(x,y)\amp\mathit{Precedes}(x,z)\to y\eqclose z]\)

This fact can be proved with the help of a kind of converse to theEquinumerosity Lemma:

Equinumerosity Lemma ‘Converse’:
\(F^{-x}\apprxclose G^{-y} \amp Fx \amp Gy \towide F\apprxclose G\)

We leave the proof of the Equinumerosity Lemma ‘Converse’and the proof of Predecessor is a Functional Relation as exercises forthe reader.

5.4 The Principle of Mathematical Induction

Let us say that a concept \(F\) ishereditary on the naturalnumbers just in case every ‘adjacent’ pair of numbers\(n\) and \(m\) (\(n\) preceding \(m\)) is such that \(m\) falls under\(F\) whenever \(n\) falls under \(F\), i.e.,

\(\mathit{HerOn}(F,N) \eqabbr \forall n\forallm[\mathit{Precedes}(n,m)\to (Fn \to Fm)]\)

Then we may state the Principle of Mathematical Induction as follows:if (a) \(0\) falls under \(F\) and (b) \(F\) is hereditary on thenatural numbers, then every natural number falls under \(F\). Informal terms:

Theorem 4:Principle of MathematicalInduction:
\(F0 \amp \mathit{HerOn}(F,N) \to \forall n Fn\)

Frege actually proves the Principle of Mathematical Induction from amore general principle that governs any \(R\)-series whatsoever. Wewill call the latter the General Principle of Induction. It assertsthat whenever a falls under \(F\), and \(F\) is hereditary on the\(R\)-series beginning with \(a\), then every member of that\(R\)-series falls under \(F\). We can formalize the General Principleof Induction with the help of a strict understanding of‘hereditary on the \(R\)-series beginning with \(a\)’.Here is a definition:

In other words, \(F\) is hereditary on the members of the \(R\)-seriesbeginning with \(a\) just in case every adjacent pair \(x\) and \(y\)in this series (with \(x\) bearing \(R\) to \(y\)) is such that \(y\)falls under \(F\) whenever \(x\) falls under \(F\). Now given thisdefinition, we can reformulate the General Principle of Induction morestrictly as:

General Principle of Induction:
\([Fa \amp \mathit{HerOn}(F, {}^{a}R^{+})] ~\to~ \forall x[R^{+}(a,x)\to Fx]\)

This is a version of Frege’s Theorem 152 inGgI, §117.

We may sketch the proof strategy as follows. Assume that theantecedent of the General Principle of Induction holds for anarbitrarily chosen concept, say \(P\). That is, assume:

\(Pa \amp \mathit{HerOn}(P, {}^{a}R^{+})\)

Now to show \(\forall x(R^{+}(a,x) \to Px)\), pick an arbitraryobject, say \(b\), and further assume \(R^{+}(a,b)\). We then simplyhave to show \(Pb\). We do this by invoking Fact (7) about \(R^{+}\)(in our subsection on the Weak Ancestral in §4). Recall that Fact(7) is:

\([Fx \amp R^{+}(x,y) \amp \mathit{Her}(F,R)] \to Fy\)

This is a theorem of logic containing the free variables \(x\), \(y\),and \(F\). First, we instantiate \(x\) and \(y\) to \(a\) and \(b\),respectively. Then, we instantiate \(F\) to the concept \([\lambda z\, R^{+}(a,z) \amp Pz]\) and apply \(\lambda\)-Conversion (thoughFrege could simply use his Rule of Substitution to achieve the sameinference). The concept being instantiated for \(F\) is the conceptmember of the R-series beginning with a and which falls underP. The result of instantiating the free variables in Fact (7) andthen applying \(\lambda\)-Conversion yields a rather long conditional,with numerous conjuncts in the antecedent and the claim that \(Pb\) inthe consequent. Thus, if the antecedent can be established, the proofis done. For those following along with pencil and paper, all of theconjuncts in the antecedent are things we already know, with theexception of the claim that \([\lambda z \, R^{+}(a,z)\amp Pz]\) ishereditary on \(R\). However, this claim can be establishedstraightforwardly from things we know to be true (and, in particular,from facts contained in the antecedent of the Principle we are tryingto prove, which we assumed as part of our conditional proof). Thereader is encouraged to complete the proof as an exercise. For thosewho would like to check their work, we give the complete Proof of theGeneral Principle of Induction here:

Proof of the General Principle of Induction

Now to derive Principle of Mathematical Induction from the GeneralPrinciple of Induction, we formulate an instance of the latter bysetting \(a\) to \(0\) and \(R\) toPrecedes:

\({[}F0 \amp \mathit{HerOn}(F, {}^{0}\mathit{Precedes}^{+}){]} \towide\forall x{[}\mathit{Precedes}^{+}(0,x) \to Fx{]}\)

When we expand the defined notation for \(\mathit{HerOn}\), substitutethe notation \(Nx\) and \(Ny\) for \(\mathit{Precedes}^{+}(0,x)\) and\(\mathit{Precedes}^{+}(0,y)\), respectively, and then employ ourrestricted quantifiers \(\forall n(\ldots n\ldots)\) and \(\forallm(\ldots m\ldots)\) for the claims of the form \(\forall y(Ny \to\ldots y\ldots)\) and \(\forall x(Nx \to \ldots x\ldots)\),respectively, the result is the Principle of Mathematical Induction(in which the notation \(\mathit{HerOn}(F,N)\) has been eliminated interms of its definiens).

5.5 Every Natural Number Has a Successor

Frege uses the Principle of Mathematical Induction to prove that everynatural number has a successor that’s a natural number. We mayformulate the theorem as follows:

Theorem 5:
\(\forall x{[}Nx \to \exists y(Ny \amp \mathit{Precedes}(x,y)){]}\)

To reconstruct Frege’s strategy for proving this theorem, recallthat the weak ancestral of the Predecessor relation, i.e.,\(\mathit{Precedes}^{+}(x,y)\), can be read as: \(x\) is a member ofthe predecessor-series ending with \(y\). Frege then considers theconceptmember of the predecessor-series ending with n, i.e.,\([\lambda z \, \mathit{Precedes}^{+}(z,n)]\), where \(n\) is anatural number. Frege then shows, by induction, that every naturalnumber \(n\) precedes the number of the conceptmember of thepredecessor-series ending with n. That is, Frege proves thatevery natural number has a successor by proving the following Lemma onSuccessors by induction:

Lemma on Successors:
\(\forall n \mathit{Precedes}(n,\#[\lambda z \,\mathit{Precedes}^{+}(z,n)])\)

This asserts that every natural number \(n\) precedes the number ofnumbers in the predecessor series ending with \(n\). Frege canestablish Theorem 5 by proving the Lemma on Successors and by showingthat the successor of a natural number is itself a natural number.

To see an intuitive picture of why the Lemma on Successors gives uswhat we want, we may temporarily regard \(\mathit{Precedes}^{+}\) asthe relation ≤. (One can prove that \(\mathit{Precedes}^{+}\) hasthe properties that ≤ has on the natural numbers.) Although wehaven’t yet assigned any meaning to the numerals ‘1’and ‘2’, the following intuitive sequence is drivingFrege’s strategy:

For example, the third member of this sequence is true because thereare 3 natural numbers (0, 1, and 2) that are less than or equal to 2;so the number 2 precedes the number of numbers less than or equal to2. Frege’s strategy is to show that the general claim, that\(n\) precedes the number of numbers less than or equal to \(n\),holds for every natural number. So, given this intuitive understandingof the Lemma on Successors, Frege has a good strategy for proving thatevery number has a successor. (For the remainder of this subsection,the reader may wish to continue to think of \(\mathit{Precedes}^{+}\)in terms of \(\leq\).)

Now to prove the Lemma on Successors by induction, we need toreconfigure this Lemma to a form which can be used as the consequentof the Principle of Mathematical Induction; i.e., we need something ofthe form \(\forall n\, Fn\). We can get the Lemma on Successors intothis form by ‘abstracting out’ a concept from the Lemmausing the right-to-left direction of \(\lambda\)-Conversion (i.e.,\(\lambda\)-Abstraction) to produce the following equivalent statementof the Lemma:

\(\forall n [\lambda y \, \mathit{Precedes}(y, \#[\lambda z \,\mathit{Precedes}^{+}(z,y)])]n\)

The concept ‘abstracted out’ is the following:

\([\lambda y\, \mathit{Precedes}(y,\#[\lambda z \,\mathit{Precedes}^{+}(z,y)])]\)

This is the concept:being an object \(y\)which precedesthe number of the concept: member of the predecessor series endingin \(y\). Let us abbreviate the \(\lambda\)-expression thatdenotes this concept as ‘\(Q\)’. Our strategy is toinstantiate the variable \(F\) in the Principle of MathematicalInduction to \(Q\). The result is therefore something that has beenproved and that we therefore know to be true:

\(Q0 \amp \mathit{HerOn}(Q,N) \to \forall nQn\)

Since the consequent is the reconfigured Lemma on Successors, Fregecan prove this Lemma by proving both that \(0\) falls under \(Q\) (cf.Gg I, Theorem 154) and that \(Q\) is hereditary onthe natural numbers (cf.Gg I, Theorem 150):

Proof that \(0\) falls under \(Q\)

Proof that \(Q\) is hereditary on the natural numbers

Given this proof of the Lemma on Successors, Theorem 5 is not faraway. The Lemma on Successors shows that every number precedes somecardinal number of the form \(\#F\). We still have to show that suchsuccessor cardinals are natural numbers. That is, it still remains tobe shown that if a number \(n\) precedes something \(y\), then \(y\)is a natural number:

Successors of Natural Numbers are Natural Numbers:
\(\forall n\forall y (\mathit{Precedes}(n,y) \to Ny)\)

Proof: Suppose that \(\mathit{Precedes}(n,a)\). Then, bydefinition, since \(n\) is a natural number,\(\mathit{Precedes}^{+}(0,n)\). So by Fact (3) about \(R^{+}\) (in thesubsection on the Weak Ancestral in §4), it follows that\(\mathit{Precedes}^*(0,a)\), and so by the definition of\(\mathit{Precedes}^{+}\), it follows that\(\mathit{Precedes}^{+}(0,a)\); i.e., \(a\) is a natural number.

Theorem 5 now follows from the Lemma on Successors and the fact thatsuccessors of natural numbers are natural numbers. With the proof ofTheorem 5, we have completed the proof of Frege’s Theorem.Before we turn to the last section of this entry, it is worthmentioning the mathematical significance of this theorem.

5.6 Arithmetic

From Frege’s Theorem, one can derive arithmetic. It is animmediate consequence Theorem 5 and the fact that Predecessor is afunctional relation that every number has aunique successor.That means we can define the successor function by adding definitedescriptions of the form ‘the \(x\) such that \(\phi\)’ toour language:

\(n' \eqdef\) the \(x\) such that \(\mathit{Precedes}(n,x)\)

We may then define the sequence of natural numbers succeeding \(0\) asfollows:

Moreover, the recursive definition of addition can now be given:

We may also officially define:

These definitions constitute the foundations of arithmetic. Frege hasthus insightfully derived the basic laws of arithmetic fromHume’s Principle in second-order logic. (Readers interested inhow these results are affected when Hume’s Principle is combinedwithpredicative second-order logic should consult Linnebo2004.)

6. Philosophical Questions Surrounding Frege’s Theorem

As we’ve now seen, the proof of Frege’s Theorem can be carriedout independently of the portion of Frege’s system which led toinconsistency. Frege himself never identified “Frege’sTheorem” as a “result”. As previously noted, heattempted to derive Hume’s Principle from Basic Law V inGg, but once the contradiction became known to him,he never officially retreated to the ‘fall-back’ positionof claiming that the proof of the Dedekind-Peano axioms fromHume’s Principle alone constituted an important result. One ofseveral reasons why he didn’t adopt this fall-back position isthat he didn’t regard Hume’s Principle as a sufficientlygeneral principle – he didn’t believe it was strongenough, from an epistemological point of view, to help us answer thequestion, “How are numbers given to us?”. We discuss thethinking behind this attitude, and other things, in what follows.

A discussion of the philosophical questions surrounding Frege’sTheorem should begin with some statement of how Frege conceived of hisown project when writingBegr,Gl,andGg. It seems clear that epistemologicalconsiderations in part motivated Frege’s work on the foundationsof mathematics. It is well documented that Frege had the followinggoal, namely, to explain our knowledge of the basic laws of arithmeticby giving an answer to the question “How are numbers‘given’ to us?” without making an appeal to thefaculty of intuition. If Frege could show that the basic laws ofnumber theory are derivable from analytic truths of logic, then hecould argue that we need only appeal to the faculty of understanding(as opposed to some faculty of intuition) to explain our knowledge ofthe truths of arithmetic. Frege’s goal then stands in contrastto the Kantian view of the exact mathematical sciences, according towhich general principles of reasoning must be supplemented by afaculty of intuition if we are to achieve mathematical knowledge. TheKantian model here is that of geometry; Kant thought that ourintuitions of figures and constructions played an essential role inthe demonstrations of geometrical theorems. (In Frege’s owntime, the achievements of Frege’s contemporaries Pasch (1882),Peano (1889b), Pieri (1898), and Hilbert (1899) showed that suchintuitions were not essential.)

6.1 Frege’s Goals and Strategy in His Own Words

Frege’s strategy then was to show that no appeal to intuition isrequired for the derivation of the theorems of number theory. This inturn required that he show that the latter are derivable using onlyrules of inference, axioms, and definitions that are purely analyticprinciples of logic. This view has become known as‘Logicism’. Here is what Frege says:

[Begr, Preface, p. 5:]
To prevent anything intuitive from penetrating here unnoticed, I hadto bend every effort to keep the chain of inferences free of gaps.[from the Bauer-Mengelberg translation in van Heijenoort 1967]

[Begr, Part III, §23:]
Through the present example, moreover, we see how pure thought,irrespective of any content given by the senses or even by anintuitiona priori, can, solely from the content that resultsfrom its own constitution, bring forth judgements that at first sightappear to be possible only on the basis of some intuition. \(\ldots\)The propositions about sequences [\(R\)-series] in what follows farsurpass in generality all those that can be derived from any intuitionof sequences. [from the Bauer-Mengelberg translation in van Heijenoort1967]

[Gl, §62:]
How, then, are numbers to be given to us, if we cannot have any ideasor intuitions of them? Since it is only in the context of aproposition that words have any meaning, our problem becomes this: Todefine the sense of a proposition in which a number word occurs. [fromthe Austin translation in Frege 1953]

[Gl, §87:]
I hope I may claim in the present work to have made it probable thatthe laws of arithmetic are analytic judgements and consequently apriori. Arithmetic thus becomes simply a development of logic, andevery proposition of arithmetic a law of logic, albeit a derivativeone. [from the Austin translation in Frege 1953]

[Gg I, §0:]
In myGrundlagen der Arithmetik, I sought to make itplausible that arithmetic is a branch of logic and need not borrow anyground of proof whatever from either experience or intuition. In thepresent book, this shall be confirmed, by the derivation of thesimplest laws of Numbers by logical means alone. [from the Furthtranslation in Frege 1967]

[Gg II, Appendix:]
The prime problem of arithmetic is the question, In what way are we toconceive logical objects, in particular, numbers? By what means are wejustified in recognizing numbers as objects? Even if this problem isnot solved to the degree I thought it was when I wrote this volume,still I do not doubt that the way to the solution has been found.[from the Furth translation in Frege 1967]

6.2 The Basic Problem for Frege’s Strategy

The basic problem for Frege’s strategy, however, is that for hislogicist project to succeed, his system must at some point include(either as an axiom or theorem) statements that explicitly assert theexistence of certain kinds of abstract entities and it is not obvioushow to justify the claim that we know such explicit existentialstatements. Given the above discussion, it should be clear that Fregeat some point inGg endorsed existence claims, either directlyin his formalism or in his metalanguage, for the following entities:

• concepts (more generally, functions)
• extensions (more generally, courses-of-value or value-ranges)
• truth-values
• numbers

Although Frege attempted to reduce the latter two kinds of entities(truth-values and numbers) to extensions, the fact is that theexistence of concepts and extensions are derivable from his Rule ofSubstitution and Basic Law V, respectively.

In light of these existence claims, a Kantian might well suggest notonly that explicit existence claims are synthetic rather than analytic(i.e., aren’t true in virtue of the meanings of the wordsinvolved) but also that since the Rule of Substitution and Basic Law Vimply existence claims, Frege cannot claim that such principles arepurely analytic principles of logic. If the Kantian is right, thensome other faculty (such as intuition) might still be needed toaccount for our knowledge of the existence claims of arithmetic.

6.3 The Existence of Concepts

Boolos (1985) noted that the Rule of Substitution causes a problem ofthis kind for Frege’s program given that it is equivalent theComprehension Principle for Concepts. Boolos suggests a defense forFrege with respect to this particular aspect of his logic, namely, toreinterpret (by paraphrasing) the second-order quantifiers so as toavoid commitment to concepts. (See Boolos 1985 for the details.)Boolos’s suggestion, however, is one which would require Fregeto abandon his realist theory of concepts. Moreover, althoughBoolos’s suggestion might lead us to an epistemologicaljustification of the Comprehension Principle for Concepts, itdoesn’t do the same for the Comprehension Principle forRelations, for his reinterpretation of the quantifiers works only forthe ‘monadic’ quantifiers (i.e., those ranging overconcepts having one argument) and thus doesn’t offer aparaphrase for quantification over relational concepts.

Another problem for a strategy of the type suggested by Boolos is thatif the second-order quantifiers are interpreted so that they do notrange over a separate domain of entities, then there is nothingappropriate to serve as the denotations of \(\lambda\)-expressions.Although Frege wouldn’t quite put it this way, ourreconstruction suggests that Frege treats open formulas with freeobject variables as if they denoted concepts. And though Fregedoesn’t use \(\lambda\)-notation, the use of such notation seemsto be the most logically perspicuous way of reconstructing his work.The use of such notation faces the same epistemological puzzles thatFrege’s Rule of Substitution faces.

To see why, note that the Principle of \(\lambda\)-Conversion:

\(\forall y([\lambda x\, \phi ]y \equiv \phi^y_x)\)

seems to be an analytic truth of logic. It says this:

An object \(y\) exemplifies the complex propertybeing suchthat \(\phi\) if and only if \(y\) is such that \(\phi\).

One might argue that this is true in virtue of the very meaning of the\(\lambda\)-expression, the meaning of \(\equiv\), and the meaning ofthe statement \([\lambda x\, \phi]y\) (which has the form \( Fx)\).However, \(\lambda\)-Conversion also implies the ComprehensionPrinciple for Concepts, for the latter follows from the former byexistential generalization:

\(\exists F\forall y(Fy \equiv \phi^y_x)\)

The point here is that the fact that an existential claim is derivablecasts at least some doubt on the purely analytic status of\(\lambda\)-Conversion. The question of how we obtain knowledge ofsuch principles is still an open question in philosophy. It is animportant question to address, since Frege’s most insightfuldefinitions are cast using quantifiers ranging over concepts andrelations (e.g., the ancestrals of a relation) and it would be usefulto have a philosophical explanation of how such entities and theprinciples which govern them become known to us. In contemporaryphilosophy, this question is still poignant, since many philosophersdo accept thatproperties andrelations of varioussorts exist. These entities are the contemporary analogues ofFrege’s concepts.

6.4 The Existence of Extensions

Though the existence of extensions falls right out of the theory ofidentity (§2.3) once terms of the form \(\epsilon F\) are addedto second-order logic, the existence of extensions that are correlatedone-to-one with concepts is a consequence of Basic Law V. The questionfor Frege’s project, then, is why should we accept as a law oflogic a statement that implies the existence of individuals and acorrelation of this kind? Frege recognized that Basic Law V’sstatus as a logical law could be doubted:

[Gg I, Preface, p. 3:]
A dispute can arise, so far as I can see, only with regard to my BasicLaw concerning courses-of-values (V) \(\ldots\) I hold that it is alaw of pure logic. [from the Furth translation in Frege 1967]

Moreover, he thought that an appeal to extensions would answer one ofthe questions that motivated his work:

[Letter to Russell, July 28, 1902:]
I myself was long reluctant to recognize ranges of values and henceclasses [sets]; but I saw no other possibility of placing arithmeticon a logical foundation. But the question is, How do we apprehendlogical objects? And I have found no other answer to it than this, Weapprehend them as extensions of concepts, or more generally, as rangesof values of functions. [from the Kaal translation in Frege 1980]

Now it is unclear why Frege thought that he could answer the questionposed here by saying that we apprehend numbers as the extensions ofconcepts. He seems to think we can answer the obvious next question“How do we apprehend extensions?” by saying “by wayof Basic Law V”. His idea here seems to be that since Basic Law V is supposed to bepurely analytic or true in virtue of the meanings of its terms, weapprehend a pair of extensions whenever we truly judge that concepts\(F\) and \(G\) are materially equivalent. Some philosophers do arguethat certainconsistent principles having the same logical form as BasicLaw V are analytic, and that such principles justifyreference to the entities described in the left-sidecondition by grounding such reference in thetruth of theright-side condition.^[15]

Why did Frege think that Basic Law V is analytic and that the materialequivalence of concepts \(F\) and \(G\) is analytically equivalent toan identity that implies the existence of extensions? To hold thatBasic Law V is analytic, it seems that one must hold that theright-side condition implies the corresponding left-side condition asa matter of meaning.^[16] This view, however, can be questioned. Suppose the right handcondition implies the left-side condition as a matter of meaning. Thatis, suppose that (R) implies (L) as a matter of meaning:

Now note that (L) itself can be analyzed, from a logical point ofview. The expression ‘\(\epsilon F\)’, though constructedfrom a term-forming operator, is really a definite description(‘the extension of \(F\)’) and so, usingRussell’s theory of descriptions, (L) can be logically analyzedas the claim:

There is an object \(x\) and an object \(y\) such that:
(1) \(x\) is a unique extension of \(F\),
(2) \(y\) is a unique extension of \(G\), and
(3) \(x = y\).

That is, for some defined or primitive notion\(\mathit{Extension}(x,F)\) (‘\(x\) is an extension of\(F\)’), (L) implies the analysis (D) as a matter ofmeaning:

But if (R) implies (L) as a matter of meaning, and (L) implies (D) asa matter of meaning, then (R) implies (D) as a matter of meaning. Thisconclusion can be questioned: why should the material equivalence of\(F\) and \(G\) imply the existence claim (D) as a matter of meaning?In other words, the suggestion that Va (i.e., the right-to-leftdirection of Basic Law V) is analytic leads to a question that has noobvious answer. Below, this line of reasoning will be adapted toquestion the analyticity of the right-to-left direction ofHume’s Principle. See Boolos 1997 (307–309), for reasonswhy \(Vb\) (the left-to-right direction of Hume’s Principle) isnot analytic.]

The moral to be drawn here is that, even if Basic Law V wereconsistent, it is not exactly clear how its right side analyticallyimplies the existence of extensions. In the end, we may need someother way of justifying our knowledge of principles like Basic Law V,that imply the existence of abstract objects – the justificationdiscussed so far seems to contain a gap. Even if we follow Frege inconceiving of extensions as ‘logical objects’, thequestion remains: how can the claims that such objects exist be trueon logical or analytic grounds alone? We might agree that there mustbe logical objects of some sort if logic is to have a subject matter,but if Frege is to achieve his goal of showing that our knowledge ofarithmetic is free of intuition, then at some point he has to addressthe question of how we can know that numbers exist. We’ll returnto this issue in the final subsection.

6.5 The Existence of Numbers and Truth-Values: The Julius Caesar Problem

Given that the proof of Frege’s Theorem makes no appeal to BasicLaw V, some philosophers have argued Frege’s best strategy forproducing an epistemologically-justified foundation for arithmetic isto replace the primitive term \(\epsilon F\) with the primitive term\(\#F\), replace Basic Law V with Hume’s Principle, and arguethat Hume’s Principle is an analytic principle of logic.^[17] However, we have just seen one reason why such a strategy does notsuffice. The claim that Hume’s Principle is an analyticprinciple of logic is subject to the same problem just posed for BasicLaw V. A reason must be given as to why the claim:

\(F \approx G\)

implies, as a matter of meaning, that:

\(\#F = \#G\)

After all, the statement “\(\#F = \#G\)” is analyzable ina manner analogous to the way we analyzed “\(\epsilon F =\epsilon G\)” in the previous section, where we usedRussell’s theory of description to analyze the sentence (L) asthe sentence (D). Following that pattern, we take the primitive notion\(\mathit{Numbers}(x,F)\) and analyze \(\#F = \#G\) as:

It is not clear why we should think that this last claim is implied by\(F \approx G\) as a matter of meaning. The right-to-left direction ofHume’s Principle is not obviously analytic.

Moreover, Frege had his own reasons for not replacing Basic Law V withHume’s Principle. One reason was that he thought Hume’sPrinciple offered no answer to the epistemological question,‘How do we grasp or apprehend logical objects, such as thenumbers?’. A second reason is that Hume’s Principle isclearly subject to ‘the Julius Caesar problem’. Fregefirst raises this problem in connection with an inductive definitionof ‘\(n = \#F\)’ that he tries out inGl,§55. Concerning this definition, Frege says:

[Gl, §56:]
… but we can never – to take a crude example —decide by means of our definitions whether any concept has the numberJulius Caesar belonging to it, or whether that conqueror of Gaul is anumber or is not. [from the Austin translation in Frege 1953]

Frege raises this same concern again for a contextual definition thatgives a ‘criterion of identity’ for the objects beingdefined. InGl §66, Frege considers thefollowing contextual definition of ‘the direction of line\(x\)’:

The direction of line \(a\) = the direction of line \(b\) if and onlyif \(a\) is parallel to \(b\).

With regard to this definition, Frege says:

[Gl, §66:]
It will not, for instance, decide for us whether England is the sameas the direction of the Earth’s axis – if I may beforgiven an example which looks nonsensical. Naturally no one is goingto confuse England with the direction of the Earth’s axis; butthat is no thanks to our definition of direction. [from the Austintranslation in Frege 1953]

Now trouble for Hume’s Principle begins to arise when werecognize that it is a contextual definition that has the same logicalform as this definition for directions. It is central to Frege’sview that the numbers areobjects, and so he believes that itis incumbent upon him to saywhich objects they are. But the‘Julius Caesar problem’ is that Hume’s Principle, ifconsidered as the sole principle offering identity conditions fornumbers, doesn’t describe the conditions under which anarbitrary object, say Julius Caesar, is or is not to be identifiedwith the number of planets. That is, Hume’s Principledoesn’t define the condition ‘\(\#F = x\)’, forarbitrary \(x\). It only offers identity conditions when \(x\) is anobject known to be a cardinal number (for then \(x = \#G\), for some\(G\), and Hume’s Principle tells us when \(\#F = \#G\)).

InGl, Frege solves the problem by giving hisexplicit definition of numbers in terms of extensions. (We describedthis in §4 above.) Unfortunately, this is only a stopgap measure,for when Frege later systematizes extensions inGg,Basic Law V has the same logical form as Hume’s Principle andthe above contextual definition of directions. Frege is aware that theJulius Caesar problem affects Basic Law V, as the discussion inGg I, §10 shows. In that section, he says(remembering that for Frege, \(\epsilon\) binds object variables andis not a function term):

[Gg I, §10:]
\(\ldots\) this by no means fixes completely the denotation of a namelike‘\(\stackrel{,}{\epsilon}\!\Phi(\epsilon)\)’.We have only a means of always recognizing a course-of-values if it isdesignated by a name like‘\(\stackrel{,}{\epsilon}\!\Phi(\epsilon)\)’, by which it is already recognizable as acourse-of-values. But, we can neither decide, so far, whether anobject is a course-of-values that is not given us as such \(\ldots\)[from the Furth translation in Frege 1967]

In other words, Basic Law V does not tell us the conditions underwhich an arbitrarily chosen object \(x\) may be identified with somegiven extension, such as \(\epsilon F\).

Until recently, it was thought that Frege solved this problem in§10 by restricting the universal quantifier \(\forall x\) of hisGg system so that it ranges only over extensions. IfFrege could have successfully restricted this quantifier toextensions, then when the question arises, whether (arbitrarilychosen) object \(x\) is identical with \(\epsilon F\), one couldanswer that \(x\) has to be the extension of some concept, say \( G\),and that Basic Law V would then tell you the conditions under which\(x\) is identical to \(\epsilon F\). On this interpretation of§10, Frege is alleged to have restricted the quantifiers when heidentified the two truth values (The True and The False) with the twoextensions that contain just these objects as members, respectively.By doing this, it was thought that all of the objects in the range ofhis quantifier \(\forall x\) inGg become extensionswhich have been identified as such, for the truth values were the onlytwo objects of his system that had not been introduced as extensionsor courses of value.

However, recent work by Wehmeier (1999) suggests that, in §10,Frege was not attempting to restrict the quantifiers of his system toextensions (nor, more generally, to courses-of-values). The extensivefootnote to §10 indicates that Frege considered, but did not holdmuch hope of, identifying every object in the domain with theextension consisting of just that object.^[18] But, more importantly, Frege later considers cases (inGg, Sections 34 and 35) which seem to presuppose thatthe domain contains objects which aren’t extensions. (In thesesections, Frege considers what happens to the definition of ‘\(x\) is a member of \(y\)’ when \(y\) is not an extension.)^[19]

Even if Frege somehow could have successfully restricted thequantifiers ofGg to avoid the Julius Caesar problem,he would no longer have been able to apply his system by extending itto include names of ordinary non-logical objects. For if he were toattempt to do so, the question, “Under what conditions is\(\epsilon F\) identical with Julius Caesar?”, would then belegitimate but have no answer. That means his logical system could notbe used for the analysis of ordinary language. But it was just theanalysis of ordinary language that led Frege to his insight that astatement of number is an assertion about a concept.

6.6 Final Observations

Even when we replace the inconsistent Basic Law V with the powerfulHume’s Principle, Frege’s work still leaves two questionsunanswered: (1) How do we know that numbers exist?, and (2) How do weprecisely specify which objects they are? The first question arisesbecause Hume’s Principle doesn’t seem to be a purelyanalytic truth of logic; if neither Hume’s Principle nor theexistential claim that numbers exist is analytically true, by whatfaculty do we come to know (the truth of) the existential claim? Thesecond question arises because the Julius Caesar problem applies toHume’s Principle; without a solution to that problem, Frege can’tclaim to have precisely specified which objects the numbers are, so asto delineate them within the domain of all logical and non-logicalobjects? So questions about the very existence and identity of numbersstill affect Frege’s work.

These two questions arise because of a limitation in the logical formof these Fregean biconditional principles such as Hume’sPrinciple and Basic Law V. These contextual definitions combine twojobs which modern logicians now typically accomplish with separateprinciples. A properly reformulated theory of ‘logical’objects should have separate principles: (1) one or more principleswhich assert theexistence of logical objects, and (2) aseparate identity principle which asserts the conditions under whichlogical objects areidentical. The latter should specifyidentity conditions for logical objects in terms of their most salientcharacteristic, one which distinguishes them from other objects. Suchan identity principle would then be more specific than the globalidentity principle for all objects (Leibniz’s Law) which assertsthat if objects \(x\) and \(y\) fall under the same concepts, they areidentical.

By way of example, consider modern set theory. Zermelo set theory \((Z)\) has several distinctive set existence principles. For example,consider the well-known Subset (or Separation) Axiom:

Subset (Separation) Axiom of Z:
\(\forall x[\mathit{Set}(x) \to \exists y[\mathit{Set}(y) \amp \forallz(z \in y \equiv (z \in x \amp \phi))]]\),
where \(\phi\) is any formula in which \(y\) isn’t free

The Subset Axiom and the other set existence axioms in Z are distinctfrom Z’s identity principle for sets:

Identity Principle for Sets:
\(\mathit{Set}(x) \amp \mathit{Set}(y) \to [\forall z(z \in x \equiv z\in y) \to x\eqclose y]\)

Note that the second principle offers identity conditions in terms ofthe most salient features of sets, namely, the fact that they, unlikeother objects, have members. The identity conditions for objects whicharen’t sets, then, can be the standard principle thatidentifies objects whenever they fall under the same concepts. Thisleads us naturally to a very general principle of identity for anyobjects whatever:

Now, if something is given to usas a set and we ask whetherit is identical with an arbitrarily chosen object \(x\), thisspecifies a clear condition that settles the matter. The onlyquestions that remain for the theory \(Z\) concern its existenceprinciple: Do we know that the Subset Axiom and other set existenceprinciples are true, and if so, how? The question of existence is thuslaid bare. We do not approach it by attempting to justify a principlethat implies the existence of sets via definite descriptions which wedon’t yet know to be well-defined.

In some classic essays (1987 and 1986/87), Boolos appears to recommendthis very procedure of using separate existence and identityprinciples. In those essays, he eschews the primitive mathematicalrelation of set membership and suggests that Frege could formulate histheory of numbers (‘Frege Arithmetic’) by using a singlenonlogical comprehension axiom which employs a specialinstantiation relation that holds between a concept \(G\) and anobject \(x\) whenever, intuitively, \(x\) is an extension consistingsolely of concepts and \(G\) is a concept ‘in’ \(x\). Hecalls this nonlogical axiom ‘Numbers’ and uses thenotation ‘\(Gηx\)’ to signify that \(G\) is in\(x\):

Numbers:
\(\forall F\exists !x\forall G(Gηx \equiv G\approx F)\)

[See Boolos 1987 (5), 1986/87 (140).] This principle asserts that for anyconcept \(F\), there is a unique object which contains in it all andonly those concepts \(G\) which are equinumerous to \(F\). Boolos thenmakes two observations: (1) Frege can then define \(\#F\) as“the unique object \(x\) such that for all concepts \(G\), \(G\)is in \(x\) iff \(G\) is equinumerous to \(F\)”, and (2)Hume’s Principle is derivable from Numbers. [See Boolos 1986/87(140).] Given these observations, we know from our work in§§4 and 5 above that Numbers suffices for the derivation ofthe basic laws of arithmetic.

Since Boolos calls this principle ‘Numbers’, it is nostretch to suppose that he would accept the following reformulation(in which ‘\(\mathit{Number}(x)\)’ is an undefined,primitive notion):

Numbers:
\(\forall F\exists !x [\mathit{Number}(x) \amp \forall G(Gηx\equiv G\approx F)]\)

Though Boolos doesn’t explicitly formulate an identity principleto complement Numbers, it seems clear that the following principlewould offer identity conditions in terms of the most distinctivefeature of numbers:

Identity Principle for Numbers:
\(\mathit{Number}(x) \amp \mathit{Number}(y) \to [\forall G(Gηx\equiv Gηy) \to x\eqclose y]\)

It is then straightforward to formulate a general principle ofidentity, as we did in the case of the set theory \(Z\):

This formulation of Frege Arithmetic, in terms of Numbers and theGeneral Principle of Identity, puts the Julius Caesar problem(described above) into better perspective; the condition ‘\(\#F= x\)’ is defined for arbitrary concepts \(F\) and objects\(x\). It openly faces the epistemological questions head-on: Do weknow that Numbers is true, and if so, how? This is where philosophersneed to concentrate their energies. [For a reconstruction of FregeArithmetic with a more general version of the special instantiationrelation η, see Zalta 1999.]

By replacing Fregean biconditionals such as Hume’s Principlewith separate existence and identity principles, we reduce twoproblems to one and and isolate the real problem for Fregeanfoundations of arithmetic, namely, the problem of giving anepistemological justification for distinctive existence claims (e.g.,Numbers) for abstract objects of a certain kind. For if anything likeFrege’s program is to succeed, it must at some point assert (asan axiom or theorem) the existence of (logical) objects of some kind.Those separate existence claims should be the focus of attention.

Bibliography

Primary Literature

Cited Works by Frege

• 1879,Begriffsschrift, eine der arithmetischen nachgebildeteFormelsprache des reinen Denkens, Halle a. S.: Louis Nebert;translation by S. Bauer Mengelberg asConcept Notation: A formulalanguage of pure thought, modelled upon that of arithmetic, in J.van Heijenoort,From Frege to Gödel: A Sourcebook inMathematical Logic, 1879–1931, Cambridge, MA: HarvardUniversity Press
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Cited Primary Works by Others

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• Die Grundlagen der Arithmetik, original German text, at Google Books.
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Related Entries

Frege, Gottlob |Frege, Gottlob: controversy with Hilbert |Frege, Gottlob: logic |Principia Mathematica |Russell, Bertrand |Russell’s paradox

Acknowledgments

I was motivated to write the present entry after reading an earlydraft of an essay by William Demopoulos. (The draft was eventuallypublished as Demopoulos and Clark 2005.) Demopoulos kindly allowed meto quote certain passages from that early draft in the footnotes tothe present entry. I am also indebted to Roberto Torretti, whocarefully read this piece and identified numerous infelicities; toFranz Fritsche, who noticed a quantifier transposition error in Fact 2about the strong ancestral; to Seyed N. Mousavian, who noticed sometypographical errors in some formulas; to Xu Mingming, who noticedthat Fact 8 about the Weak Ancestral (Section 4, subsection “TheWeak Ancestral of \(R\)”) was missing an important condition(namely, that \(R\) must be 1–1); to Evgeni Latinov, who notedthat the discussion in Section 2.7 (of how the Russell paradox isengendered in Frege’s system) also requires that the materialequivalence of \(F\) and \(G\) be a sufficient for the identity of\(F\) and \(G\); and to Paul Pietroski, who noticed an infelicity inthe first statement of the principle of induction in Section 4. I amindebted to Kai Wehmeier, who (a) reminded me that, strictly speaking,the result of replacing Basic Law V by Hume’s Principle inFrege’s system does not result in a subsystem of the originaluntil we replace the primitive notion “the course of values ofthe function \(f\)” with the primitive notion “the numberof \(F\)s”, and (b) refereed the July 2013 update to this entryand developed numerous, insightful suggestions forimprovement. Finally, I am indebted to Jerzy Hanusek for pointing outthat Existence of Extensions principle can be derived more simply inFrege’s system directly from the classical logic ofidentity, and to Rowena Conway for noticing some unnoticed corruption caused by our new MathJax rendering of the formulas.