Internal Set Theory - Intuitive Justification

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 numbers, 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 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.