2,454 research outputs found
Nonstandard proof methods in toposes
We determine sufficient structure for an elementary topos to emulate E.
Nelson's Internal Set Theory in its internal language, and show that any topos
satisfying the internal axiom of choice occurs as a universe of standard
objects and maps. This development allows one to employ the proof methods of
nonstandard analysis (transfer, standardisation, and idealisation) in new
environments such as toposes of -sets and Boolean \'etendues
A theory of hyperfinite sets
We develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS,
which is based on the idea of existence of proper subclasses of big finite
sets. We demonstrate how theorems of classical continuous mathematics can be
transfered to THS, prove consistency of THS and present some applications.Comment: 28 page
Reverse Mathematics and parameter-free Transfer
Recently, conservative extensions of Peano and Heyting arithmetic in the
spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been
proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis
restricted to formulas without parameters. Based on this axiom, we formulate a
base theory for the Reverse Mathematics of Nonstandard Analysis and prove some
natural reversals, and show that most of these equivalences do not hold in the
absence of parameter-free Transfer.Comment: 22 pages; to appear in Annals of Pure and Applied Logi
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