Internal set theory
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Internal set theory (IST) is a mathematical theory of sets developed by
Edward Nelson Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematical ...
that provides an axiomatic basis for a portion of the
nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
introduced by
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorpo ...
. Instead of adding new elements to the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, 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 conventional ZFC axioms for sets. Thus, IST is an enrichment of ZFC: 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 of
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
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 is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
that were initially required to justify rigorously the consistency of number systems containing 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 Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, \sqrt, ca ...
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 is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite sets) are ac ...
. * 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 term ''standard'' 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 of ''nonstandard'' 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). * 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 is by definition merely another standard number. * Nevertheless, there are (many) nonstandard whole numbers in any infinite subset of N. * 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 a
logically consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is 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 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.


Formal axioms for IST

IST is an axiomatic theory in the
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
with equality in a
language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
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 :\begin\exists^\mathrmx\,\phi(x)&=\exists x\,(\operatorname(x)\land\phi(x)),\\ \forall^\mathrmx\,\phi(x)&=\forall x\,(\operatorname(x)\to\phi(x)).\end IST includes all axioms of the
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
with the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
(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 initial in its name: Idealisation, Standardisation, and Transfer.


''I'': Idealisation

*For any internal formula \phi without a free occurrence of ''z'', the universal closure of the following formula is an axiom: *:\forall^\mathrmz\,(z\text\to\exists y\,\forall x\in z\,\phi(x,y,u_1,\dots,u_n))\leftrightarrow\exists y\,\forall^\mathrmx\,\phi(x,y,u_1,\dots,u_n). *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 (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.


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 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. If ''S'' is nonstandard, 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 nonstandard 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 nonstandard set contains a nonstandard 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 \phi is any formula (it may be external) without a free occurrence of ''y'', the universal closure of *:\forall^\mathrmx\,\exists^\mathrmy\,\forall^\mathrmt\,(t\in y\leftrightarrow(t\in x\land\phi(t,u_1,\dots,u_n))) :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 nonstandard elements is not prescribed).


T: Transfer

*If \phi(x,u_1,\dots,u_n) is an internal formula with no other free variables than those indicated, then *:\forall^\mathrmu_1\dots\forall^\mathrmu_n\,(\forall^\mathrmx\,\phi(x,u_1,\dots,u_n)\to\forall x\,\phi(x,u_1,\dots,u_n)) :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 ''x''s as soon as it holds for all standard ''x''s—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 Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
,
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating L ...
,
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, Augustin-Louis Cauchy, and others were the reason that they were originally abandoned for the more cumbersome
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
-based arguments developed by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
,
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
, and
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, 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 A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
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
elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
by noting they can be modeled by an appropriate interpretation of
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
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 In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a superthe ...
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.Nelson, Edward (1977). Internal set theory: A new approach to nonstandard analysis.
Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...
83(6):1165–1198.


Related theories

Related theories were developed by
Karel Hrbacek Karel may refer to: People * Karel (given name) * Karel (surname) * Charles Karel Bouley, talk radio personality known on air as Karel * Christiaan Karel Appel, Dutch painter Business * Karel Electronics, a Turkish electronics manufacturer * Gr ...
and others.


Notes


References

* Robert, Alain (1985). ''Nonstandard analysis''. John Wiley & Sons. .
Internal Set Theory
a chapter of an unfinished book by Nelson. {{Infinitesimals Systems of set theory Nonstandard analysis