Elementary function arithmetic: Difference between revisions

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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 manner, with just the rules of substitution and induction, and defining equations for all elementary recursive functions. HoweverUnlike PRA, however, the elementary recursive functions can be charactized 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 ==