Implementation of mathematics in set theory: Difference between revisions

Revision as of 16:31, 2 June 2022

Latest revision as of 22:58, 2 May 2025

(3 intermediate revisions by 3 users not shown)

Line 1:

{{Short description|none}}

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]]).

Line 65 ⟶ 66:

The '''downward closure''' of a member <math>x</math> of the field of <math>R</math> is the smallest set <math>D</math> containing <math>x</math>, and containing each <math>zRy</math> for each <math>y \in D</math> (i.e., including the preimage of each of its elements with respect to <math>R</math> as a subset.)

The '''[[relation composition|relative product]]''' <math>R|;S</math> of <math>R</math> and <math>S</math> is the relation <math>\left\{\left(x, z\right) : \exists y\,\left(xRy \wedge ySz\right)\right\}</math>.

Notice that with our formal definition of a binary relation, the range and codomain of a relation are not distinguished. This could be done by representing a relation <math>R</math> with codomain <math>B</math> as <math>\left(R, B\right)</math>, but our development will not require this.

Line 103 ⟶ 104:

=== Special kinds of function ===

A function is an '''[[injective]]''' (also called '''[[bijection|one-to-one]]''') if it has an inverse function.

A function <math>f</math> from <math>A</math> to <math>B</math> is a:

* '''[[Injective function|Injection]]''' from <math>A</math> to <math>B</math> if the [[image (mathematics)|image]]s under <math>f</math> of distinct members of <math>A</math> are distinct members of <math>B</math>.

Line 319 ⟶ 318:

*Tourlakis, George, 2003. ''Lectures in Logic and Set Theory, Vol. 2''. Cambridge Univ. Press.

== External links ==

* [http://us.metamath.org/ Metamath:] A web site devoted to an ongoing derivation of mathematics from the axioms of ZFC and [[first-order logic]].

* [[Stanford Encyclopedia of Philosophy]]:

** [http://plato.stanford.edu/entries/quine-nf Quine's New Foundations]—by Thomas Forster.

Line 326:

* Randall Holmes: [https://randall-holmes.github.io/nf.html New Foundations Home Page]

{{Mathematical logic}}

[[Category:Large-scale mathematical formalization projects]]

[[Category:Formalism (deductive)]]

[[Category:Mathematical logic]]

[[Category:Set theory]]

[[Category:Formalism (deductive)]]

[[Category:Large-scale mathematical formalization projects]]