Foundational Aspects of "non"standard Mathematics
David Ballard
jan. 1994 · Contemporary mathematics - American Mathematical Society Bog 176 · American Mathematical Soc.
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This work proposes a major new extension of non standard mathematics. Addressed to a general mathematical audience, the book is intended to be philosophically provocative. The model theory on which non standard mathematics has been based is first reformulated within point set topology, which facilitates proofs and adds perspective. These topological techniques are then used to give new, uniform conservativity proofs for the various versions of non standard mathematics proposed by Nelson, Hrbacek and Kawai. The proofs allow for sharp comparison. Addressing broader issues, Ballard then argues that what is novel in these forms of non standard mathematics is the introduction, however tentative, of relativity in one's mathematical environment. This hints at the possibility of a mathematical environment which is radically relativistic. The work's major and final feature is to present and prove conservative a version of non standard mathematics which, for the first time, illustrates this full radical relativism. The book is entirely self-contained, with all necessary background in point set topology, model theory, non standard analysis, and set theory provided in full.
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