About: Implementation of mathematics in set theory

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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 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 Infinity and Choice).

Attributes	Values
rdfs:label	Implementation of mathematics in set theory (en)
rdfs:comment	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 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 Infinity and Choice). (en)
sameAs	Implementation of mathematics in set theory Implementation of mathematics in set theory
dbp:wikiPageUsesTemplate	dbt:Main_article
Subject	Set theory Formalism (deductive) Mathematical logic Large-scale mathematical formalization projects
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Implementation_of_mathematics_in_set_theory?oldid=1043356355&ns=0
Wikipage page ID	3486002 (xsd:integer)
page length (characters) of wiki page	60725 (xsd:nonNegativeInteger)
Wikipage revision ID	1043356355 (xsd:integer)
Link from a Wikipage to another Wikipage	Formal language Norbert Wiener Projection (mathematics) Ronald Jensen Union (set theory) Well-formed formula Well-founded relation Well-order Antisymmetric relation Axiom of choice Axiom of infinity Axiom of pairing Axiom of union Axiom schema of replacement Axiom schema of specification Bertrand Russell Bijection Binary relation Up to Urelement Von Neumann universe Von Neumann–Bernays–Gödel set theory Cantor's paradox Cardinal number Cartesian product Set theory Free variables and bound variables Formalism (deductive) Mathematical logic Set-builder notation Set-theoretic definition of natural numbers Set theory Surjective function Zermelo set theory Zermelo–Fraenkel set theory Class (set theory) Dana Scott Predicate (mathematical logic) Principia Mathematica Composition of relations Converse relation Ordered pair Ordinal number Partially ordered set Empty set Equinumerosity Equivalence class Equivalence relation First-order logic Function (mathematics) Composition operator Image (mathematics) Kazimierz Kuratowski Keith Devlin Cantor–Bernstein theorem Scott's trick Symmetric relation Large-scale mathematical formalization projects Injective function Inverse function John von Neumann Peano axioms Transfinite induction Transitive relation Stanford Encyclopedia of Philosophy Stratification (mathematics) Theory of descriptions Total order Large cardinal Measurable cardinal History of numbers New Foundations Reflexive relation Relation (mathematics) dbr:Mahlo_cardinals dbr:Mirimanoff's_paradox
has abstract	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 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 Infinity and 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 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. It is not the primary aim of this article to say anything about the relative merits of these theories as foundations for mathematics. The reason for the use of two different set theories is to illustrate that multiple approaches to the implementation of mathematics are feasible. Precisely because of this approach, this article is not a source of "official" definitions for any mathematical concept. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Implementation_of_mathematics_in_set_theory
is Wikipage redirect of	Formalized mathematics Mathematical formalization
is Link from a Wikipage to another Wikipage of	Formalism (philosophy of mathematics) Foundations of mathematics Journal of Formalized Reasoning Functional programming Mizar system Interactive Theorem Proving (conference) New Foundations

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