It is often claimed that analysis with infinitesimals requires more substantial use of the Axiom Choice than traditional elementary analysis. The claim based on observation hyperreals entail existence nonprincipal ultrafilters over N, a strong version Choice, while real numbers can be constructed in ZF. axiomatic approach to nonstandard methods refutes this objection. We formulate theory SPOT s...

A recent debate has focused on different methodological principles underlying the practice of axiom choice in mathematics (cf. Feferman et al., 2000; Maddy, 1997; Easwaran, 2008). The general aim of these contributions can be described as twofold: first to clarify the spectrum of informal justification strategies retraceable in the history of mathematical axiomatics. Second, to evaluate and to ...

§9.1 The Axiom of Choice We come now to the most important part of set theory for other branches of mathematics. Although infinite set theory is technically the foundation for all mathematics, in practice it is perfectly valid for a mathematician to ignore it – with two exceptions. Of course most of the basic set constructions as outlined in chapter 2 (unions, intersections, cartesian products,...