Internal Set Theory Intuitive Justification Principles Of The

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.

1 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.

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.

1.1 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.

2 Formal axioms for IST

There are three axioms of IST to add to the established ZFC set theoretic axioms - 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.

2.1 I : Idealisation

For each standard, finite set F, if there exists a classical relation R such that R( g, f ) holds for all f in F and some g, then there is a particular G with R( G, f ) for any standard f.

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

2.1.1 Applied to the relation ≠

More precisely we take for the relation R ( g, f ) : g and f are in infinite set S but are not equal. Since "For every standard, finite subset F of infinite set 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 infinite set 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).

2.1.2 Applied to the relation <

Since "For every standard, finite set of natural numbers F there is a 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 G such that G > f for all standard f." In other words, there exists a natural number greater than any standard number.

2.1.3 Applied to the relation ∈

More precisely we take for R ( g, f ) : g is a finite subset of S containing element f. Since "For every standard, finite subset F of S, 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 G such that f ∈ G for all standard f in S." In other words, all standard elements of S can be contained in a finite subset of S.

2.2 S : Standardisation

If A is a standard set and P any property, classical or otherwise, then there is 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).

2.3 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 x's.

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

3 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 Weierstrasse.

The approach for internal set theory is the same as that for any new axiomatic - 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: anything (correct or incorrect!) that can be proven within internal set theory can be proven in the Zermelo-Fraenkel axioms alone.

4 External links and resources

Robert, Alain (1985). NonStandard Analysis John Wiley & Sons. BooksEnthsiast.com

Set theorySet theory is the mathematical theory of sets, which represent collections of abstract objects. It has a central role in modern mathematical theory, providing the basic language in which most of mathematics is expressed. For more information on set theory

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