Implementation of mathematics in set theory: Difference between revisions

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<ref></ref>This article examines the implementation of mathematical concepts in [[set theory]]. The implementation of a number of basic mathematical concepts is carried out in parallel in [[ZFC]] (the dominant set theory) and in [[New Foundations|NFU]], the version of Quine's [[New Foundations]] shown to be consistent by [[R. B. Jensen]] in 1969 (here understood to include at least axioms of [[Axiom of infinity|Infinity]] and [[Axiom of choice|Choice]]).

What is said here applies also to two families of set theories: on the one hand, a range of theories including [[Zermelo set theory]] near the lower end of the scale and going up to ZFC extended with [[large cardinal property|large cardinal]] hypotheses such as "there is a [[measurable cardinal]]"; and on the other hand a hierarchy of extensions of NFU which is surveyed in the [[New Foundations]] article. These correspond to different general views of what the set-theoretical universe is like, and it is the approaches to implementation of mathematical concepts under these two general views that are being compared and contrasted.