Natural restrictions on the syntax of the second-order (i.e., polymorphic) lambda calculus are of interest for programming language theory. One of the authors showed in Leivant (1991) that when type abstraction in that calculus is stratified into levels, the definable numeric functions are precisely the super-elementary functions (level [Escr ]4 in the Grzegorczyk Hierarchy). We define here a second-order lambda calculus in which type abstraction is stratified to levels up to ωω, an ordinal that permits highly uniform (and finite) type inference rules. Referring to this system, we show that the numeric functions definable in the calculus using ranks < ω[lscr ] are precisely Grzegorczyk's class [Escr ][lscr ]+3 ([lscr ] [ges ] 1). This generalizes Leivant (1991), where this is proved for [lscr ] = 1. Thus, the numeric functions definable in our calculus are precisely the primitive recursive functions.
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