Internal set theory
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Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson which provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, the axioms introduce a new term, "standard", which can be used to make discriminations not possible under the conventional axioms for sets. In particular, non-standard elements within the set of real numbers can be shown to have properties that correspond to the properties of infinitesimal and unlimited elements.

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

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Intuitive justification

Whilst IST has a perfectly formal axiomatic scheme, described below, an intuitive justification of the meaning of the term 'standard' is desirable. This is not part of the official theory, but is a pedagogical device that might help the student engage with the formalism. The essential distinction, similar to the concept of definable numbers, contrasts the finiteness of the domain of concepts that we can specify and discuss with the unbounded infinity of the set of numbers.

  • The number of symbols we write 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) civilisation.
  • Hence there is only a finite set of whole numbers our civilisation can discuss in its allotted timespan.
  • 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 the fact that there is such a limit, whilst the set of whole numbers continues forever without bound, is a mathematical truth.

The term standard is therefore intuitively taken to correspond to some necessarily finite portion of "accessible" whole numbers. In fact the argument can be applied to any infinite set of objects whatsoever - there are only so many elements that we 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 of non-standard elements too large or too anonymous to grasp within any infinite set.

Principles of the standard predicate

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 which does not use the new predicate standard explicitly or implicitly will be termed a Classical Formula.
  • Any definition which does so is, of course, termed a Non-Classical Formula.
  • Any number uniquely specified by a classical formula is standard (by definition).
  • The non-standard numbers are precisely those which cannot be uniquely specified (due to limitations of time and space) by a classical formula.
  • Non-standard 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 is by-definition merely another standard number.
  • Nevertheless, there are (many) non-standard whole numbers in any infinite subset of N.
  • Non-standard numbers are completely ordinary numbers, having decimal representations, prime factorisations, etc. Every classical theorem that applies to the natural numbers applies to the non-standard 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 a classical formula that uniquely defined a non-standard number.
  • The predicate "non-standard" is a logically consistent method for distinguishing large numbers - the usual term will be illimited. 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 non-standard 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 non-standard set.

Formal axioms for IST

There are three axioms of IST to add to the established ZFC set theoretic axioms (note that use of the ZFC axiom schemas is restricted: the axiom schemas of separation and replacement can only be used with classical formulas, just as in ZFC proper) - conveniently one for each letter in the name: Idealisation, Standardisation, and Transfer. All the principles described above can be formally derived from these three additional axiom schemes.

I : Idealisation

  • For every classical relation R, and for arbitrary values for all other free variables, we have that if for each standard, finite set F, there exists a g such that R( g, f ) holds for all f in F, then there is a particular G such that for any standard f we have R( G, f ), and conversely, if there exists G such that for any standard f, we have R( G, f ), then for each finite set F, there exists a g such that R( g, f ) holds for all f in F.

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

Applied to the relation ≠

If S is standard and finite, we take for the relation R ( g , f ) : g and f are in S but are not equal. Since the intersection of two standard finite sets is standard (by Transfer - see below) and finite, and since "For every standard, finite subset F of S there is an element g in S such that g ≠ f for all f in F." is false (since no such g exists in the case where F = S), then we may use Idealisation to tell us that "There is a G in S such that G ≠ f for all standard f in S " is also false, i.e. all the elements of S are standard.

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

If S is infinite, then we take for the relation R ( g, f ) : g and f are in S but are not equal. Since "For every standard, finite subset F of S there is an element g in S such that g ≠ f for all f in F." - say by choosing g as any element of S not in F - we may use Idealisation to derive "There is a G in S such that G ≠ f for all standard f in S ." In other words, every infinite set contains a non-standard element (many, in fact).

If S is non-standard, we take for the relation R ( g, f ) : g and f are in S but are not equal. Since "For every standard, finite subset F of S there is an element g in S such that g ≠ f for all f in F." - say by choosing g as any element of S not in F (F cannot be equal to S since F is standard and S is non-standard) - we may use Idealisation to derive "There is a G in S such that G ≠ f for all standard f in S ." In other words, every non-standard set contains a non-standard element.

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

Applied to the relation <

Since "For every standard, finite set of natural numbers F there is a natural number g such that g > f for all f in F." - say, g = maximum( F ) + 1 - we may use Idealisation to derive "There is a natural number G such that G > f for all standard natural numbers f." In other words, there exists a natural number greater than any standard natural number.

Applied to the relation ∈

More precisely we take for R ( g, f ) : g is a finite set containing element f. Since "For every standard, finite set F, there is a finite set g such that f ∈ g for all f in F." - say by choosing g = F itself - we may use Idealisation to derive "There is a finite set G such that f ∈ G for all standard f." For any set S, the intersection of S with the set G is a finite subset of S which contains every standard element of S.

S : Standardisation

  • If A is a standard set and P any property, classical or otherwise, then there is a unique, standard subset B of A whose standard elements are precisely the standard elements of A satisfying P (but the behaviour of B's non-standard elements is not prescribed).

T : Transfer

  • If all the parameters A, B, C, ..., W of a classical formula F have standard values then F( x, A, B,..., W ) holds for all x's as soon as it holds for all standard xs.

From which it follows that all uniquely defined concepts or objects within classical mathematics are standard.

Formal justification for the axioms

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 were the reason that they were originally abandoned for the more cautious, but rigorous, limit-based arguments developed by Cauchy and Karl Weierstrass.

The approach for internal set theory is the same as that for any new axiomatic system - we construct a model 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 of non-Euclidean geometry by noting they can be modeled by an appropriate interpretation of great circles on a sphere in ordinary 3-space.

In fact via a suitable model a proof can be given of the relative consistency of ZFC + IST as compared with ZFC: if ZFC is consistent, then ZFC + IST is consistent. In fact, a stronger statement can be made: ZFC + IST is a conservative extension of ZFC: any classical formula (correct or incorrect!) that can be proven within internal set theory can be proven in the Zermelo-Fraenkel axioms with the Axiom of Choice alone.

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