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- Second-order_arithmetic abstract "In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. It was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik. The standard axiomatization of second-order arithmetic is denoted Z2.Second-order arithmetic includes, but is significantly stronger than, its first-order counterpart Peano arithmetic. Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of numbers as well as numbers themselves. Because real numbers can be represented as (infinite) sets of natural numbers in well-known ways, and because second order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic. For this reason, second-order arithmetic is sometimes called “analysis”.Second-order arithmetic can also be seen as a weak version of set theory in which every element is either a natural number or a set of natural numbers. Although it is much weaker than Zermelo-Fraenkel set theory, second-order arithmetic can prove essentially all of the results of classical mathematics expressible in its language. A subsystem of second-order arithmetic is a theory in the language of second-order arithmetic each axiom of which is a theorem of full second-order arithmetic (Z2). Such subsystems are essential to reverse mathematics, a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength. Much of core mathematics can be formalized in these weak subsystems, some of which are defined below. Reverse mathematics also clarifies the extent and manner in which classical mathematics is nonconstructive.".
- Second-order_arithmetic wikiPageExternalLink 2272259.
- Second-order_arithmetic wikiPageExternalLink sosoa.
- Second-order_arithmetic wikiPageExternalLink Proofs%2BTypes.html.
- Second-order_arithmetic wikiPageID "3542454".
- Second-order_arithmetic wikiPageRevisionID "591339950".
- Second-order_arithmetic hasPhotoCollection Second-order_arithmetic.
- Second-order_arithmetic subject Category:Formal_theories_of_arithmetic.
- Second-order_arithmetic type Abstraction100002137.
- Second-order_arithmetic type Cognition100023271.
- Second-order_arithmetic type Explanation105793000.
- Second-order_arithmetic type FormalTheoriesOfArithmetic.
- Second-order_arithmetic type HigherCognitiveProcess105770664.
- Second-order_arithmetic type Process105701363.
- Second-order_arithmetic type PsychologicalFeature100023100.
- Second-order_arithmetic type Theory105989479.
- Second-order_arithmetic type Thinking105770926.
- Second-order_arithmetic comment "In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. It was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik.".
- Second-order_arithmetic label "Second-order arithmetic".
- Second-order_arithmetic sameAs m.09kh4g.
- Second-order_arithmetic sameAs Q7442973.
- Second-order_arithmetic sameAs Q7442973.
- Second-order_arithmetic sameAs Second-order_arithmetic.
- Second-order_arithmetic wasDerivedFrom Second-order_arithmetic?oldid=591339950.
- Second-order_arithmetic isPrimaryTopicOf Second-order_arithmetic.