Internal set theory (IST) is a mathematical theory ofsets developed byEdward Nelson that provides an axiomatic basis for a portion of thenonstandard analysis introduced byAbraham Robinson. Instead of adding new elements to thereal numbers, Nelson's approach modifies the axiomatic foundations through syntactic enrichment. Thus, the axioms introduce a new term, "standard", which can be used to make discriminations not possible under the conventionalZFC axioms for sets. Thus, IST is an enrichment ofZFC: all axioms of ZFC are satisfied for all classical predicates, while the new unary predicate "standard" satisfies three additional axioms I, S, and T. In particular, suitable nonstandard elements within the set of real numbers can be shown to have properties that correspond to the properties ofinfinitesimal and unlimited elements.

Nelson's formulation is made more accessible for the lay-mathematician by leaving out many of the complexities of meta-mathematicallogic that were initially required to justify rigorously the consistency of number systems containing infinitesimal elements.

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Whilst IST has a perfectly formal axiomatic scheme, described below, an intuitive justification of the meaning of the termstandard is desirable. This isnot part of the formal theory, but is a pedagogical device that might help the student interpret the formalism. The essential distinction, similar to the concept ofdefinable numbers, contrasts the finiteness of the domain of concepts that we can specify and discuss, with the unbounded infinity of the set of numbers; comparefinitism.

• The number of symbols one writes with is finite.
• The number of mathematical symbols on any given page is finite.
• The number of pages of mathematics a single mathematician can produce in a lifetime is finite.
• Any workable mathematical definition is necessarily finite.
• There are only a finite number of distinct objects a mathematician can define in a lifetime.
• There will only be a finite number of mathematicians in the course of our (presumably finite) civilization.
• Hence there is only a finite set of whole numbers our civilization can discuss in its allotted lifespan.
• What that limit actually is, is unknowable to us, being contingent on many accidental cultural factors.
• This limitation is not in itself susceptible to mathematical scrutiny, but that there is such a limit, whilst the set of whole numbers continues forever without bound, is a mathematical truth.

The termstandard is therefore intuitively taken to correspond to some necessarily finite portion of "accessible" whole numbers. The argument can be applied to any infinite set of objects whatsoever – there are only so many elements that one can specify in finite time using a finite set of symbols and there are always those that lie beyond the limits of our patience and endurance, no matter how we persevere. We must admit to a profusion ofnonstandard elements—too large or too anonymous to grasp—within any infinite set.

The following principles follow from the above intuitive motivation and so should be deducible from the formal axioms. For the moment we take the domain of discussion as being the familiar set of whole numbers.

• Any mathematical expression that does not use the new predicatestandard explicitly or implicitly is aninternal formula.
• Any definition that does so is anexternal formula.
• Any numberuniquely specified by an internal formula is standard (by definition).
• Nonstandard numbers are precisely those that cannot be uniquely specified (due to limitations of time and space) by an internal formula.
• Nonstandard numbers are elusive: each one is too enormous to be manageable in decimal notation or any other representation, explicit or implicit, no matter how ingenious your notation. Whatever you succeed in producing isby definition merely another standard number.
• Nevertheless, there are (many) nonstandard whole numbers in any infinite subset ofN.
• Nonstandard numbers are completely ordinary numbers, having decimal representations, prime factorizations, etc. Every classical theorem that applies to the natural numbers applies to the nonstandard natural numbers. We have created, not new numbers, but a new method of discriminating between existing numbers.
• Moreover, any classical theorem that is true for all standard numbers is necessarily true for all natural numbers. Otherwise the formulation "the smallest number that fails to satisfy the theorem" would be an internal formula that uniquely defined a nonstandard number.
• The predicate "nonstandard" is alogically consistent method for distinguishinglarge numbers—the usual term will beillimited. Reciprocals of these illimited numbers will necessarily be extremely small real numbers—infinitesimals. To avoid confusion with other interpretations of these words, in newer articles on IST those words are replaced with the constructs "i-large" and "i-small".
• There are necessarily only finitely many standard numbers—but caution is required: we cannot gather them together and hold that the result is a well-defined mathematical set. This will not be supported by the formalism (the intuitive justification being that the precise bounds of this set vary with time and history). In particular we will not be able to talk about the largest standard number, or the smallest nonstandard number. It will be valid to talk about some finite set that contains all standard numbers—but this non-classical formulation could only apply to a nonstandard set.

IST is an axiomatic theory in thefirst-order logic with equality in alanguage containing a binary predicate symbol ∈ and a unary predicate symbol st(x). Formulas not involving st (i.e., formulas of the usual language of set theory) are called internal, other formulas are called external. We use the abbreviations

IST includes all axioms of theZermelo–Fraenkel set theory with theaxiom of choice (ZFC). Note that the ZFC schemata ofseparation andreplacement arenot extended to the new language, they can only be used with internal formulas. Moreover, IST includes three new axiom schemata – conveniently one for each initial in its name:Idealisation,Standardisation, andTransfer.

• For any internal formula${\displaystyle \varphi }$ without a free occurrence ofz, the universal closure of the following formula is an axiom:

  ${\displaystyle {\mathrm{\forall }}^{\mathrm{s}\mathrm{t}}z(z\text{ is finite}\to \mathrm{\exists }y\mathrm{\forall }x\in z\varphi (x,y,u_1,\dots ,u_n))\leftrightarrow \mathrm{\exists }y{\mathrm{\forall }}^{\mathrm{s}\mathrm{t}}x\varphi (x,y,u_1,\dots ,u_n).}$

• In words: For every internal relationR, and for arbitrary values for all other free variables, we have that if for each standard, finite setF, there exists ag such that⁠${\displaystyle R(g,f)}$⁠ holds for allf inF, then there is a particularG such that forany standardf we have⁠${\displaystyle R(g,f)}$⁠, and conversely, if there existsG such that for any standardf, we have⁠${\displaystyle R(g,f)}$⁠, then for each finite setF, there exists ag such that⁠${\displaystyle R(g,f)}$⁠ holds for allf inF.

The statement of this axiom comprises two implications. The right-to-left implication can be reformulated by the simple statement that elements of standard finite sets are standard. The more important left-to-right implication expresses that the collection of all standard sets is contained in a finite (nonstandard) set, and moreover, this finite set can be taken to satisfy any given internal property shared by all standard finite sets.

This very general axiom scheme upholds the existence of "ideal" elements in appropriate circumstances. Three particular applications demonstrate important consequences.

IfS is standard and finite, we take for the relation⁠${\displaystyle R(g,f)}$⁠:g andf are not equal andg is inS. Since "For every standard finite set F there is an element g in S such that⁠${\displaystyle g\ne f}$⁠ for all f inF" is false (no suchg exists whenF =S), we may use Idealisation to tell us that "There is a G in S such that⁠${\displaystyle g\ne f}$⁠ for all standardf" is also false, i.e. all the elements ofS are standard.

IfS is infinite, then we take for the relation⁠${\displaystyle R(g,f)}$⁠:g andf are not equal andg is inS. Since "For every standard finite set F there is an element g in S such that⁠${\displaystyle g\ne f}$⁠ for all f inF" (the infinite setS is not a subset of the finite setF), we may use Idealisation to derive "There is a G in S such that⁠${\displaystyle g\ne f}$⁠ for all standardf." In other words, every infinite set contains a nonstandard element (many, in fact).

The power set of a standard finite set is standard (by Transfer) and finite, so all the subsets of a standard finite set are standard.

IfS is nonstandard, we take for the relation⁠${\displaystyle R(g,f)}$⁠:g andf are not equal andg is inS. Since "For every standard finite set F there is an element g in S such that⁠${\displaystyle g\ne f}$⁠ for all f inF" (the nonstandard setS is not a subset of the standard and finite setF), we may use Idealisation to derive "There is a G in S such that⁠${\displaystyle g\ne f}$⁠ for all standard f." In other words, every nonstandard set contains a nonstandard element.

As a consequence of all these results, all the elements of a setS are standard if and only ifS is standard and finite.

Since "For every standard, finite set of natural numbers F there is a natural number g such that⁠${\displaystyle g>f}$⁠ for all f inF" (say,g = max(F) + 1), we may use Idealisation to derive "There is a natural number G such that⁠${\displaystyle g>f}$⁠ for all standard natural numbersf." In other words, there exists a natural number greater than each standard natural number.

We take⁠${\displaystyle R(g,f)}$⁠:g is a finite set containing elementf. Since "For every standard, finite set F, there is a finite set g such that⁠${\displaystyle f\in G}$⁠ for all f inF" (e.g.g =F), we may use Idealisation to derive "There is a finite set G such that⁠${\displaystyle f\in G}$⁠ for all standardf." For any setS, the intersection ofS with the setG is a finite subset ofS that contains every standard element ofS.G is necessarily nonstandard, by the ZFCregularity axiom.

• If${\displaystyle \varphi }$ is any formula (it may be external) without a free occurrence ofy, the universal closure of

  ${\displaystyle {\mathrm{\forall }}^{\mathrm{s}\mathrm{t}}x{\mathrm{\exists }}^{\mathrm{s}\mathrm{t}}y{\mathrm{\forall }}^{\mathrm{s}\mathrm{t}}t(t\in y\leftrightarrow (t\in x\wedge \varphi (t,u_1,\dots ,u_n)))}$

is an axiom.

• In words: IfA is a standard set and P any property, internal or otherwise, then there is a unique, standard subsetB ofA whose standard elements are precisely the standard elements ofA satisfyingP (but the behaviour ofB's nonstandard elements is not prescribed).

• If${\displaystyle \varphi (x,u_1,\dots ,u_n)}$ is an internal formula with no other free variables than those indicated, then

  ${\displaystyle {\mathrm{\forall }}^{\mathrm{s}\mathrm{t}}{u}_1\dots {\mathrm{\forall }}^{\mathrm{s}\mathrm{t}}{u}_n({\mathrm{\forall }}^{\mathrm{s}\mathrm{t}}x\varphi (x,u_1,\dots ,u_n)\to \mathrm{\forall }x\varphi (x,u_1,\dots ,u_n))}$

is an axiom.

• In words: If all the parametersA,B,C, ...,W of an internal formulaF have standard values thenF(x,A,B,...,W) holds for allx's as soon as it holds for all standardx's—from which it follows that all uniquely defined concepts or objects within classical mathematics are standard.

Aside from the intuitive motivations suggested above, it is necessary to justify that additional IST axioms do not lead to errors or inconsistencies in reasoning. Mistakes and philosophical weaknesses in reasoning about infinitesimal numbers in the work ofGottfried Leibniz,Johann Bernoulli,Leonhard Euler,Augustin-Louis Cauchy, and others were the reason that they were originally abandoned for the more cumbersome^[1]real number-based arguments developed byGeorg Cantor,Richard Dedekind, andKarl Weierstrass, which were perceived as being more rigorous by Weierstrass's followers.

The approach for internal set theory is the same as that for any new axiomatic system—we construct amodel for the new axioms using the elements of a simpler, more trusted, axiom scheme. This is quite similar to justifying the consistency of the axioms ofellipticnon-Euclidean geometry by noting they can be modeled by an appropriate interpretation ofgreat circles on a sphere in ordinary 3-space.

In fact via a suitable model a proof can be given of the relative consistency of IST as compared with ZFC: if ZFC is consistent, then IST is consistent. In fact, a stronger statement can be made: IST is aconservative extension of ZFC: any internal formula that can be proven within internal set theory can be proven in the Zermelo–Fraenkel axioms with the axiom of choice alone.^[2]

Related theories were developed byKarel Hrbacek and others.

• Robert, Alain (1985).Nonstandard analysis. John Wiley & Sons.ISBN 0-471-91703-6.
• Internal Set Theory, a chapter of an unfinished book by Nelson.

• Systems of set theory
• Nonstandard analysis

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