Implementation of mathematics in set theory: Difference between revisions

Revision as of 15:41, 27 August 2021

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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.

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=== Operations on functions ===

Let <math>f</math> and <math>g</math> be arbitrary functions. The '''[[function composition|composition]]''' of <math>f</math> and <math>g</math>, <math>g \circ f</math>, is defined as the relative product <math>f\,|\,g</math>, but only if this results in a function such that <math>g \circ f</math> is also a function, with <math>\left(g \circ f\right)\!\left(x\right) = g\!\left(f\!\left(x\right)\right)</math>, if the range of <math>f</math> is a subset of the ___domain of <math>g</math>. The '''[[inverse function|inverse]]''' of <math>f</math>, <math>f^\left(-1\right)</math>, is defined as the [[inverseconverse relation|converse]] of <math>f</math> if this is a function. Given any set <math>A</math>, the identity function <math>i_A</math> is the set <math>\left\{\left(x, x\right) \mid x \in A\right\}</math>, and this is a set in both ZFC and NFU for different reasons.

=== Special kinds of function ===