Countable algebra and set existence axioms
type: Item
instance of: scholarly article
title: Countable algebra and set existence axioms
zbMATH Open document ID: 0575.03038
DOI: 10.1016/0168-0072(83)90012-X
author: Harvey M. Friedman
author: Stephen G. Simpson
author: Rick L. Smith
published in: Annals of Pure and Applied Logic
publication date: 1983-01-01T00:00:00Z
review text: Which set existence axioms are needed to prove the theorems of ordinary mathematics ? This question was raised by Harvey Friedman at the ICM in 1974, and the authors turn to it here in the field of algebra. The procedure is to encode the structures (say, Abelian groups) and statements about them (say, ''every (countable) Abelian group has a torsion subgroup'') within second order number theory (hence the restriction to countable algebra). The encoding is, in this case, less artificial than in cases considered earlier (say, analysis). The authors encounter again the theme of Reverse Mathematics: there are five subsystems of second order number theory ranging, in increasing strength, from \(RCA_ 0\) (recursive comprehension), which is needed to make sense of the encoding, to \(\Pi^ 1_ 1\)-CA\({}_ 0\) \((\Pi^ 1_ 1\)- comprehension); and, if a theorem of ordinary mathematics is proved in the weakest possible subsystem, the statement of the theorem turns out to be provably equivalent to that subsystem over the next weaker subsystem. (The above statement about the torsion subgroup turns out to be equivalent to arithmetical comprehension over \(RCA_ 0.)\) In an introduction the authors explain the relationship between their work and computable algebra and constructive algebra.
Mathematics Subject Classification ID: 03F35
Mathematics Subject Classification ID: 03E30
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 03A05
zbMATH DE Number: 3918362
zbMATH Keywords: second-order arithmetic
zbMATH Keywords: encoding of structures
zbMATH Keywords: Reverse Mathematics
zbMATH Keywords: comprehension
zbMATH Keywords: weakest possible subsystem
reviewed by: Gebhard Fuhrken
MaRDI profile type: Publication
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full work available at URL: 0168-0072(83)90012-x
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description: scientific article; zbMATH DE number 3918362