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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, 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 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 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.
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