Internal set theory is a mathematical theory of sets developed by Edward Nelson

Edward Nelson

Edward Nelson is a professor in the Mathematics Department at Princeton University. He is known for his work on mathematical physics and mathematical logic...

 that provides an axiomatic basis for a portion of the non-standard analysis

Non-standard analysis

Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...

 introduced by Abraham Robinson

Abraham Robinson

Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....

. Instead of adding new elements to the real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, the axioms introduce a new term, "standard", which can be used to make discriminations not possible under the conventional axioms for sets. Thus, the starting point of IST is a modification of ZFC

Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...

. In particular, suitable non-standard elements within the set of real numbers can be shown to have properties that correspond to the properties of infinitesimal

Infinitesimal

Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

 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

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

 that were initially required to justify rigorously the consistency of infinitesimal elements.

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 formal theory, but is a pedagogical device that might help the student interpret the formalism. The essential distinction, similar to the concept of definable number

Definable number

A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ holds in the standard model of set theory .For the purposes of this article,...

s, contrasts the finiteness of the domain of concepts that we can specify and discuss with the unbounded infinity of the set of numbers; compare finitism

Finitism

In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...

.

• 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) 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 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 that does not use the new predicate standard explicitly or implicitly is an internal formula.
• Any definition that does so is an external formula.
• Any number uniquely specified by an internal formula is standard (by definition).
• Non-standard numbers are precisely those that cannot be uniquely specified (due to limitations of time and space) by an internal 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 factorizations, 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 an internal 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

IST is an axiomatic theory in the first-order logic

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

 with equality in a language

Signature (logic)

In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes.Signatures play the same...

 containing a binary predicate symbol ∈ and a unary predicate symbol standard(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 the Zermelo–Fraenkel set theory

Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...

 with the axiom of choice (ZFC). Note that the ZFC schemata of separation and replacement are not extended to the new language, they can only be used with internal formulas. Moreover, IST includes three new axiom schemata – conveniently one for each letter in its name: Idealisation, Standardisation, and Transfer.

I: Idealisation

• For any internal formula without free occurrence of z, the universal closure of the following formula is an axiom:

  

• In words: For every internal 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.

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 (non-standard) 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.

Applied to the relation ≠

If S is standard and finite, we take for the relation R(g, f): g and f are not equal and g is in S. Since "For every standard finite set F there is an element g in S such that for all f in F" is false (no such g exists when ), we may use Idealisation to tell us that "There is a G in S such that for all standard f" is also false, i.e. all the elements of S are standard.

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

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

As a consequence of all these results, all the elements of a set 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 for all f in F" – say, – we may use Idealisation to derive "There is a natural number G such that for all standard natural numbers f." In other words, there exists a natural number greater than each 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 for all f in F" – say by choosing itself – we may use Idealisation to derive "There is a finite set G such that for all standard f." For any set S, the intersection of S with the set G is a finite subset of S that contains every standard element of S. G is necessarily nonstandard.

S: Standardisation

• If is any formula (it may be external) without a free occurrence of y, the universal closure of

  

is an axiom.

• In words: If A is a standard set and P any property, internal 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 is an internal formula with no other free variables than those indicated, then

  

is an axiom.

• In words: If all the parameters A, B, C, ..., W of an internal formula F have standard values then holds for all xs 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 in the work of Gottfried Leibniz

Gottfried Leibniz

Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

, Johann Bernoulli

Johann Bernoulli

Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...

, Leonhard Euler

Leonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

, Augustin-Louis Cauchy, and others were the reason that they were originally abandoned for the more cumbersome real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

-based arguments developed by Georg Cantor

Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...

, Richard Dedekind

Richard Dedekind

Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...

, and Karl Weierstrass

Karl Weierstrass

Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....

, 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 a model

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

 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

Non-Euclidean geometry

Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

 by noting they can be modeled by an appropriate interpretation of great circle

Great circle

A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

s 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 a conservative extension

Conservative extension

In mathematical logic, a logical theory T_2 is a conservative extension of a theory T_1 if the language of T_2 extends the language of T_1; every theorem of T_1 is a theorem of T_2; and any theorem of T_2 which is in the language of T_1 is already a theorem of T_1.More generally, if Γ is a set of...

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.