Editing Internal set theory (section)

==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]]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]].
* 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 <u>by definition</u> 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]] 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.