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Line 1: 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.

Implementation of mathematics in set theory: Difference between revisions