On relative nearstandardness in ist

Abstract

M. F. Prokhorova UDC 513.83 In the article [1], E. Gordon defined the predicate "to be standard relative to t': zstt r 3't~o : (Ffin(~)) & (t E dome) & (z E ~(t)) (here F fin(~) means that ~ is a function whose values are finite sets) and introduced the natural notions of t-infinitesimal, t-illimited number, and t-limited real number. For example, a t-infinitesimal is defined as a number whose modulus is less than every positive t-standard number (this notion is a particular case of the notion of superinfinitesimal or 7r-monad [2]). It was demonstrated in [1] that, for some nonstandard natural number N, not all points of the interval [0, 1] are N-nearstandard (i.e., there exists z E [0, 1] such that there is no N-standard number N-infinitely close to x). The following question arises: is it possible to choose a nonstandard natural number N so that each point of the interval [0, 1] has an N-standard part, i.e., is N-nearstandard. If the answer to this question were positive then we could construct an external function similar to but "more detailed" than the standard part map. Unfortunately, as is shown below, the answer to this question is negative. Moreover, it remains negative in a more general case when we replace the set of natural numbers with an arbitrary set of nonmeasurable cardinality and the interval with an arbitrary Hausdorff space other than a rare compact set [3]. The theorems below are also of interest from the viewpoint of constructing the propositions of bounded internal set theory, BIST, of [4] which are equivalent to the conjecture of existence of mea- surable cardinals [5] (this conjecture is unprovable in ZFC). Let (X, r) be a topological space (r is the family of open sets). We call t X-appropriate if all points of X are t-nearstandard (i.e., for each x E X there is a t-standard y E X such that x is contained in every t-standard neighborhood of y). It is clear that such t must be bounded (i.e., t lies in some standard set). We define the complexity of a bounded t to be the cardinal complt = min{cardT : st(T) & t e T}.