Non-standard Nonstandard Analysis and the computational content of standard mathematics

Abstract

The aim of this paper is to highlight a hitherto unknown computational aspect of Nonstandard Analysis. Recently, a number of nonstandard versions of Goedel's system T have been introduced ([2,9,12]), and it was shown in [26] that the systems from [2] play a pivotal role in extracting computational information from proofs in Nonstandard Analysis. It is a natural question if similar techniques may be used to extract computational information from proofs not involving Nonstandard Analysis. In this paper, we provide a positive answer to this question using the nonstandard system from [9]. This system validates so-called non-standard uniform boundedness principles which are central to Kohlenbach's approach to proof mining ([14]). In particular, we show that from classical and ineffective existence proofs (not involving Nonstandard Analysis but using weak Koenig's lemma), one can `automatically' extract approximations to the objects claimed to exist.