Countable algebra and set existence axioms
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.
scientific article; zbMATH DE number 3918362
scientific article