Elementary function arithmetic: Difference between revisions

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In [[proof theory]], a branch of [[mathematical logic]], '''elementary function arithmetic''', also called '''EFA''', '''elementary arithmetic''' and '''exponential function arithmetic''', is the system of arithmetic with the usual elementary properties of 0, 1, +, ×, ''x''^''y'', together with [[mathematical induction|induction]] for formulas with [[bounded quantifiers]].

EFA is a very weak logical system, whose [[proof theoretic ordinal]] is ω³, but still seems able to prove much of ordinary mathematics that can be stated in the language of [[Peano axioms|first-order arithmetic]].

==Definition==

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*a binary relation symbol < (This is not really necessary as it can be written in terms of the other operations and is sometimes omitted, but is convenient for defining bounded quantifiers).

'''Bounded quantifiers''' are those of the form ∀(x<y) and ∃ (x<y) which are abbreviations for ∀ x (x<y)→,,,→… and ∃x (x<y)∧...∧… in the usual way.

The axioms of EFA are

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==Friedman's grand conjecture<!--'Friedman's grand conjecture' redirects here-->==

[[Harvey Friedman]]'s '''grand conjecture''' implies that many mathematical theorems, such as [[Fermat's lastLast theoremTheorem]], can be proved in very weak systems such as EFA.

The original statement of the conjecture from {{harvtxt|Friedman|1999}} is:

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==Related systems==

Several related computational complexity classes have similar properties to EFA:

*One can omit the binary function symbol exp from the language, by taking Robinson arithmetic together with induction for all formulas with bounded quantifiers and an axiom stating roughly that exponentiation is a function defined everywhere. This is similar to EFA and has the same proof theoretic strength, but is more cumbersome to work with.

*There are weak fragments of second-order arithmetic called RCA{{su|p=*|b=0}} and WKL{{su|p=*|b=0}} that have the same consistency strength as EFA and are conservative over it for Π{{su|p=0|b=2}} sentences{{explain}}, which are sometimes studied in [[reverse mathematics]] {{harv|Simpson|2009}}.

*'''Elementary recursive arithmetic''' ('''ERA''') is a subsystem of [[primitive recursive arithmetic]] (PRA) in which recursion is restricted to [[ELEMENTARY#Definition|bounded sums and products]]. This also has the same Π{{su|p=0|b=2}} sentences as EFA, in the sense that whenever EFA proves ∀x∃y P(x,y), with P quantifier-free, ERA proves the open formula P(x,T(x)), with T a term definable in ERA. Like PRA, ERA can be defined in an entirely logic-free{{clarify}} manner, with just the rules of substitution and induction, and defining equations for all elementary recursive functions. Unlike PRA, however, the elementary recursive functions can be characterized by the closure under composition and projection of a ''finite'' number of basis functions, and thus only a finite number of defining equations are needed.

Like PRA, ERA can be defined in an entirely logic-free manner, with just the rules of substitution and induction, and defining equations for all elementary recursive functions. Unlike PRA, however, the elementary recursive functions can be characterized by the closure under composition and projection of a ''finite'' number of basis functions, and thus only a finite number of defining equations are needed.

== See also ==

<!-- *[[ELEMENTARY]], a related computational complexity class - already wikilinked in the text as <nowiki>[[ELEMENTARY#Definition|bounded sums and products]]</nowiki> -->

*[[Grzegorczyk hierarchy]]

*[[Reverse mathematics]]