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Encyclopedia > 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 illimited elements. The word set, which is among the words with the most numerous definitions in the English language (at 464 definitions according to the Oxford English Dictionary), may have one of the following meanings. ... Reverend Edward Nelson was the father of British naval commander Horatio Nelson. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... Abraham Robinson (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 incorporated into mathematics. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...

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. Jump to: navigation, search Logic (from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy amongst philosophers (see below). ...


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 assist 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. A definable number is a real number which can be unambiguously defined by some mathematical statement. ...

  • 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.
  • 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. The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ...

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 gf 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 Gf 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 gf 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 Gf 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 gf 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 Gf 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 iff 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 fg 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 fG 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. In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Karl Weierstraß Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // The Theory of GOLDFiSH AND CHEESE!!!111oneoneone11!!!! Goldfish are like zomg soo kewl! Yea i knowww how kewl are goldfish?!! Soundness of Goldfish...

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 mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the...

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 alone. // Definition A logical theory T2 is a conservative extension of theory T1 if any consequence of T2 involving symbols of T1 only is already a consequence of T1. ...

External links and resources

  • Robert, Alain (1985). NonStandard Analysis John Wiley & Sons. ISBN 0471917036

  Results from FactBites:
Set theory Summary (2598 words)
Naive set theory is the original set theory developed by mathematicians at the end of the 19th century.
Axiomatic set theory is a rigorous axiomatic branch of mathematics developed in response to the discovery of serious flaws (such as Russell's paradox) in naïve set theory.
Internal set theory is an extension of axiomatic set theory that admits infinitesimal and illimited non-standard numbers.
Set Theory (Stanford Encyclopedia of Philosophy) (3279 words)
Set Theory is the mathematical science of the infinite.
The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics.
There are four main directions of current research in set theory, all intertwined and all aiming at the ultimate goal of the theory: to describe the structure of the mathematical universe.
  More results at FactBites »



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