In thefoundations of mathematics,Morse–Kelley set theory (MK),Kelley–Morse set theory (KM),Morse–Tarski set theory (MT),Quine–Morse set theory (QM) or thesystem of Quine and Morse is afirst-orderaxiomatic set theory that is closely related tovon Neumann–Bernays–Gödel set theory (NBG). While von Neumann–Bernays–Gödel set theory restricts thebound variables in the schematic formula appearing in theaxiom schema ofclass comprehension to range over sets alone, Morse–Kelley set theory allows these bound variables to range overproper classes as well as sets, as first suggested byQuine in 1940 for his systemML.

Morse–Kelley set theory is named after mathematiciansJohn L. Kelley andAnthony Morse and was first set out byWang (1949)^[1] and later in an appendix to Kelley's textbookGeneral Topology (1955), a graduate level introduction totopology.^{[2][non-primary source needed]} Kelley said the system in his book was a variant of the systems due toThoralf Skolem and Morse. Morse's own version appeared later in his bookA Theory of Sets (1965).

While von Neumann–Bernays–Gödel set theory is aconservative extension ofZermelo–Fraenkel set theory (ZFC, the canonical set theory) in the sense that a statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC, Morse–Kelley set theory is aproper extension of ZFC (assuming ZFC is consistent). Unlike von Neumann–Bernays–Gödel set theory, where the axiom schema of class comprehension can be replaced with finitely many of its instances, Morse–Kelley set theory cannot be finitely axiomatized.

NBG and MK share a commonontology. Theuniverse of discourse consists ofclasses. Classes that are members of other classes are calledsets. A class that is not a set is aproper class. The primitiveatomic sentences involve membership or equality.

With the exception of class comprehension, the following axioms are the same as those forNBG, inessential details aside. The symbolic versions of the axioms employ the following notational devices:

• The upper case letters other thanM, appearing in Extensionality, Class Comprehension, and Foundation, denote variables ranging over classes. A lower case letter denotes a variable that cannot be aproper class, because it appears to the left of an ∈. As MK is a one-sorted theory, this notational convention is onlymnemonic.
• Themonadicpredicate${\displaystyle Mx,}$ whose intended reading is "the classx is a set", abbreviates${\displaystyle \mathrm{\exists }W(x\in W).}$
• Theempty set${\displaystyle \varnothing }$ is defined by${\displaystyle \mathrm{\forall }x(x\notin \varnothing ).}$
• The classV, theuniversal class having all possible sets as members, is defined by${\displaystyle \mathrm{\forall }x(Mx\to x\in V).}$V is also thevon Neumann universe.

Extensionality: Classes having the same members are the same class.

${\displaystyle \mathrm{\forall }X\mathrm{\forall }Y(\mathrm{\forall }z(z\in X\leftrightarrow z\in Y)\to X=Y).}$

A set and a class having the same extension are identical. Hence MK is not a two-sorted theory, appearances to the contrary notwithstanding.

Foundation: Each nonempty classA isdisjoint from at least one of its members.

${\displaystyle \mathrm{\forall }A[A\ne \varnothing \to \mathrm{\exists }b(b\in A\wedge \mathrm{\forall }c(c\in b\to c\notin A))].}$

Class Comprehension: Let φ(x) be any formula in the language of MK in whichx is afree variable andY is not free. φ(x) may contain parameters that are either sets or proper classes. More consequentially, the quantified variables in φ(x) may range over all classes and not just over all sets;this is the only way MK differs fromNBG. Then there exists aclass${\displaystyle Y=\{x\mid \varphi (x)\}}$ whose members are exactly thosesetsx such that${\displaystyle \varphi (x)}$ comes out true. Formally, ifY is not free in φ:

${\displaystyle \mathrm{\forall }{W}_1...W_n\mathrm{\exists }Y\mathrm{\forall }x[x\in Y\leftrightarrow (\varphi (x,W_1,...W_n)\wedge Mx)].}$

Pairing: For any setsx andy, there exists a set${\displaystyle z=\{x,y\}}$ whose members are exactlyx andy.

${\displaystyle \mathrm{\forall }x\mathrm{\forall }y[(Mx\wedge My)\to \mathrm{\exists }z(Mz\wedge \mathrm{\forall }s[s\in z\leftrightarrow (s=x\vee s=y)])].}$

Pairing licenses the unordered pair in terms of which theordered pair,${\displaystyle ⟨x,y⟩}$, may be defined in the usual way, as${\displaystyle \{\{x\},\{x,y\}\}}$. With ordered pairs in hand, Class Comprehension enables definingrelations andfunctions on sets as sets of ordered pairs, making possible the next axiom:

Limitation of Size:C is aproper class if and only ifV can bemapped one-to-one intoC.

${\displaystyle \begin{array}{l}\mathrm{\forall }C[\mathrm{\neg }MC\leftrightarrow \mathrm{\exists }F(\mathrm{\forall }x[Mx\to \mathrm{\exists }s(s\in C\wedge ⟨x,s⟩\in F)]\wedge \\ \mathrm{\forall }x\mathrm{\forall }y\mathrm{\forall }s[(⟨x,s⟩\in F\wedge ⟨y,s⟩\in F)\to x=y])].\end{array}}$

The formal version of this axiom resembles theaxiom schema of replacement, and embodies the class functionF. The next section explains how Limitation of Size is stronger than the usual forms of theaxiom of choice.

Power set: Letp be a class whose members are all possiblesubsets of the seta. Thenp is a set.

${\displaystyle \mathrm{\forall }a\mathrm{\forall }p[(Ma\wedge \mathrm{\forall }x[x\in p\leftrightarrow \mathrm{\forall }y(y\in x\to y\in a)])\to Mp].}$

Union: Let${\displaystyle s=\bigcup a}$ be the sum class of the seta, namely theunion of all members ofa. Thens is a set.

${\displaystyle \mathrm{\forall }a\mathrm{\forall }s[(Ma\wedge \mathrm{\forall }x[x\in s\leftrightarrow \mathrm{\exists }y(x\in y\wedge y\in a)])\to Ms].}$

Infinity: There exists an inductive sety, meaning that (i) theempty set is a member ofy; (ii) ifx is a member ofy, then so is${\displaystyle x\cup \{x\}.}$.

${\displaystyle \mathrm{\exists }y[My\wedge \varnothing \in y\wedge \mathrm{\forall }z(z\in y\to \mathrm{\exists }x[x\in y\wedge \mathrm{\forall }w(w\in x\leftrightarrow [w=z\vee w\in z])])].}$

Note thatp ands in Power Set and Union are universally, not existentially, quantified, as Class Comprehension suffices to establish the existence ofp ands. Power Set and Union only serve to establish thatp ands cannot be proper classes.

The above axioms are shared with other set theories as follows:

• ZFC andNBG: Pairing, Power Set, Union, Infinity;
• NBG (and ZFC, if quantified variables were restricted to sets): Extensionality, Foundation;
• NBG: Limitation of Size.

Monk (1980) and Rubin (1967) are set theory texts built around MK; Rubin'sontology includesurelements. These authors and Mendelson (1997: 287) submit that MK does what is expected of a set theory while being less cumbersome thanZFC andNBG.

MK is strictly stronger than ZFC and itsconservative extension NBG, the other well-known set theory withproper classes. In fact, NBG—and hence ZFC—can be proved consistent in MK. That means that if MK's axioms hold, one can define aTrue predicate and show that all the ZFC and NBG axioms are true—hence every other statement formulated in ZFC or NBG is true, because truth is preserved by logic. MK's strength stems from its axiom schema of Class Comprehension beingimpredicative, meaning that φ(x) may contain quantified variables ranging over classes. The quantified variables in NBG's axiom schema of Class Comprehension are restricted to sets; hence Class Comprehension in NBG must bepredicative. (Separation with respect to sets is still impredicative in NBG, because the quantifiers in φ(x) may range over all sets.) The NBG axiom schema of Class Comprehension can be replaced with finitely many of its instances; this is not possible in MK. MK is consistent relative to ZFC augmented by an axiom asserting the existence of stronglyinaccessible cardinals.

The only advantage of theaxiom of limitation of size is that it implies theaxiom of global choice. Limitation of Size does not appear in Rubin (1967), Monk (1980), or Mendelson (1997). Instead, these authors invoke a usual form of the localaxiom of choice, and an "axiom of replacement,"^[3] asserting that if thedomain of a class function is a set, itsrange is also a set. Replacement can prove everything that Limitation of Size proves, except prove some form of theaxiom of choice.

Limitation of Size plusI being a set (hence the universe is nonempty) renders provable the sethood of the empty set; hence no need for anaxiom of empty set. Such an axiom could be added, of course, and minor perturbations of the above axioms would necessitate this addition. The setI is not identified with thelimit ordinal${\displaystyle \omega ,}$ asI could be a set larger than${\displaystyle \omega .}$ In this case, the existence of${\displaystyle \omega }$ would follow from either form of Limitation of Size.

The class ofvon Neumann ordinals can bewell-ordered. It cannot be a set (under pain of paradox); hence that class is a proper class, and all proper classes have the same size asV. HenceV too can be well-ordered.

MK can be confused with second-order ZFC, that is, ZFC withsecond-order logic (representing second-order objects in set rather than predicate language) as its background logic. The language of second-order ZFC is similar to that of MK (although a set and a class having the same extension can no longer be identified), and theirsyntactical resources for practical proof are almost identical (and are identical if MK includes the strong form of Limitation of Size). But thesemantics of second-order ZFC are quite different from those of MK. For example, if MK is consistent then it has a countable first-order model, while second-order ZFC has no countable models.

ZFC, NBG, and MK each have models describable in terms ofV, thevon Neumann universe of sets inZFC. Let theinaccessible cardinalκ be a member ofV. Also let Def(X) denote the Δ₀ definablesubsets ofX (seeconstructible universe). Then:

• V_κ is model ofZFC;
• Def(V_κ) is a model of Mendelson's version ofNBG, which excludes global choice, replacing limitation of size by replacement and ordinary choice;
• V_κ+1, thepower set ofV_κ, is a model of MK.

MK was first set out inWang (1949)^[1] and popularized in an appendix toJ. L. Kelley's (1955)General Topology, using the axioms given in the next section. The system of Anthony Morse's (1965)A Theory of Sets is equivalent to Kelley's, but formulated in an idiosyncratic formal language rather than, as is done here, in standardfirst-order logic. The first set theory to includeimpredicative class comprehension wasQuine'sML, that built onNew Foundations rather than onZFC.^[4]Impredicative class comprehension was also proposed inMostowski (1951) andLewis (1991).

The axioms and definitions in this section are, but for a few inessential details, taken from the Appendix to Kelley (1955). The explanatory remarks below are not his. The Appendix states 181 theorems and definitions, and warrants careful reading as an abbreviated exposition of axiomatic set theory by a working mathematician of the first rank. Kelley introduced his axioms gradually, as needed to develop the topics listed after each instance ofDevelop below.

Notations appearing below and now well-known are not defined. Peculiarities of Kelley's notation include:

• He didnot distinguish variables ranging over classes from those ranging over sets;
• domain f andrange f denote the domain and range of the functionf; this peculiarity has been carefully respected below;
• His primitive logical language includesclass abstracts of the form${\displaystyle \{x:A(x)\},}$ "the class of all setsx satisfyingA(x)."

Definition:x is aset (and hence not aproper class) if, for somey,${\displaystyle x\in y}$.

I. Extent: For eachx and eachy,x=y if and only if for eachz,${\displaystyle z\in x}$ when and only when${\displaystyle z\in y.}$

Identical toExtensionality above.I would be identical to theaxiom of extensionality inZFC, except that the scope ofI includes proper classes as well as sets.

II. Classification (schema): An axiom results if in

For each${\displaystyle \beta }$,${\displaystyle \beta \in \{\alpha :A\}}$ if and only if${\displaystyle \beta }$ is a set and${\displaystyle B,}$

'α' and 'β' are replaced by variables, 'A ' by a formula Æ, and 'B ' by the formula obtained from Æ by replacing each occurrence of the variable that replaced α by the variable that replaced β provided that the variable that replaced β does not appear bound inA.

Develop: Booleanalgebra of sets. Existence of thenull class and of the universal classV.

III. Subsets: Ifx is a set, there exists a sety such that for eachz, if${\displaystyle z\subseteq x}$, then${\displaystyle z\in y.}$

The import ofIII is that ofPower Set above. Sketch of the proof of Power Set fromIII: for anyclassz that is a subclass of the setx, the classz is a member of the sety whose existenceIII asserts. Hencez is a set.

Develop:V is not a set. Existence ofsingletons.Separation provable.

IV. Union: Ifx andy are both sets, then${\displaystyle x\cup y}$ is a set.

The import ofIV is that ofPairing above. Sketch of the proof of Pairing fromIV: the singleton${\displaystyle \{x\}}$ of a setx is a set because it is a subclass of the power set ofx (by two applications ofIII). ThenIV implies that${\displaystyle \{x,y\}}$ is a set ifx andy are sets.

Develop: Unordered andordered pairs,relations,functions,domain,range,function composition.

V. Substitution: Iff is a [class] function anddomain f is a set, thenrange f is a set.

The import ofV is that of theaxiom schema of replacement inNBG andZFC.

VI. Amalgamation: Ifx is a set, then${\displaystyle \bigcup x}$ is a set.

The import ofVI is that ofUnion above.IV andVI may be combined into one axiom.^[5]

Develop:Cartesian product,injection,surjection,bijection,order theory.

VII. Regularity: If${\displaystyle x\ne \varnothing }$ there is a membery ofx such that${\displaystyle x\cap y=\varnothing .}$

The import ofVII is that ofFoundation above.

Develop:Ordinal numbers,transfinite induction.

VIII. Infinity: There exists a sety, such that${\displaystyle \varnothing \in y}$ and${\displaystyle x\cup \{x\}\in y}$ whenever${\displaystyle x\in y.}$

This axiom, or equivalents thereto, are included in ZFC and NBG.VIII asserts the unconditional existence of two sets, theinfinite inductive sety, and the null set${\displaystyle \varnothing .}$${\displaystyle \varnothing }$ is a set simply because it is a member ofy. Up to this point, everything that has been proved to exist is a class, and Kelley's discussion of sets was entirely hypothetical.

Develop:Natural numbers,N is a set,Peano axioms,integers,rational numbers,real numbers.

Definition:c is achoice function ifc is a function and${\displaystyle c(x)\in x}$ for each memberx ofdomain c.

IX. Choice: There exists a choice functionc whose domain is${\displaystyle V-\{\varnothing \}.}$

IX is very similar to theaxiom of global choice derivable fromLimitation of Size above.

Develop:Equivalents of the axiom of choice. As is the case withZFC, the development of thecardinal numbers requires some form of choice.

If the scope of all quantified variables in the above axioms is restricted to sets, all axioms exceptIII and the schemaIV are ZFC axioms.IV is provable in ZFC. Hence the Kelley treatment ofMK makes very clear that all that distinguishesMK from ZFC are variables ranging overproper classes as well as sets, and the Classification schema.

• Kelley, John L. (1975) [1955].General Topology.Graduate Texts in Mathematics. Vol. 27 (2nd ed.). New York: Springer-Verlag.ISBN 978-0-387-90125-1.OCLC 1365153.
• Lemmon, E. J. (1986)Introduction to Axiomatic Set Theory. Routledge & Kegan Paul.
• David K. Lewis (1991)Parts of Classes. Oxford: Basil Blackwell.
• Mendelson, Elliott (1987).Introduction to Mathematical Logic. Chapman & Hall.ISBN 0-534-06624-0. The definitive treatment of the closely related set theoryNBG, followed by a page on MK. Harder than Monk or Rubin.
• Monk, J. Donald (1980)Introduction to Set Theory. Krieger. Easier and less thorough than Rubin.
• Morse, A. P., (1965)A Theory of Sets. Academic Press.
• Mostowski, Andrzej (1950),"Some impredicative definitions in the axiomatic set theory"(PDF),Fundamenta Mathematicae,37:111–124,doi:10.4064/fm-37-1-111-124.
• Rubin, Jean E. (1967)Set Theory for the Mathematician. San Francisco: Holden Day. More thorough than Monk; the ontology includesurelements.

• DownloadGeneral Topology (1955) by John L. Kelley in various formats. The appendix contains Kelley's axiomatic development of MK.

From Foundations of Mathematics (FOM) discussion group:

• Allen Hazen on set theory with classes.
• Joseph Shoenfield's doubts about MK.

• Systems of set theory

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