Elementary function arithmetic: Difference between revisions

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{{redirect|Elementary recursive arithmetic|the computational complexity class|ELEMENTARY}}

In [[proof theory]], a branch of [[mathematical logic]], '''elementary function arithmetic''' ('''EFA'''), also called '''elementary arithmetic''' and '''exponential function arithmetic''',<ref>C. Smoryński, "Nonstandard Models and Related Developments" (p. 217). From ''Harvey Friedman's Research on the Foundations of Mathematics'' (1985), Studies in Logic and the Foundations of Mathematics vol. 117.</ref> is the system of arithmetic with the usual elementary properties of 0,&nbsp;1,&nbsp;+,&nbsp;×,&nbsp;''x''<supmath>''x^y''</supmath>, together with [[mathematical induction|induction]] for formulas with [[bounded quantifier]]s.

EFA is a very weak logical system, whose [[proof theoretic ordinal]] is ω<supmath>\omega^3</supmath>, but still seems able to prove much of ordinary mathematics that can be stated in the language of [[Peano axioms|first-order arithmetic]].

==Definition==