Definition
EFA is a system in first order logic (with equality). Its language contains: *two constants 0, 1, *three binary operations +, ×, exp, with exp(''x'',''y'') usually written as ''x''''y'', *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 and which are abbreviations for and in the usual way. The axioms of EFA are *The axioms of Robinson arithmetic for 0, 1, +, ×, < *The axioms for exponentiation: ''x''0 = 1, ''x''''y''+1 = ''x''''y'' × ''x''. *Induction for formulas all of whose quantifiers are bounded (but which may contain free variables).Friedman's grand conjecture
Harvey Friedman's grand conjecture implies that many mathematical theorems, such as Fermat's Last Theorem, can be proved in very weak systems such as EFA. The original statement of the conjecture from is: : "Every theorem published in the '' Annals of Mathematics'' whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. EFA is the weak fragment ofRelated 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 and WKL that have the same consistency strength as EFA and are conservative over it for Π sentences, which are sometimes studied in reverse mathematics . *Elementary recursive arithmetic (ERA) is a subsystem of primitive recursive arithmetic (PRA) in which recursion is restricted to bounded sums and products. This also has the same Π 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 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
* * * * *References
* * * {{Mathematical logic Conjectures Formal theories of arithmetic Proof theory