Provably Total Functions of Arithmetic with Basic Terms
نویسنده
چکیده
This paper presents a new characterization of provably recursive functions of first-order arithmetic. We consider functions defined by sets of equations. The equations can be arbitrary, not necessarily defining primitive recursive, or even total, functions. The main result states that a function is provably recursive iff its totality is provable (using natural deduction) from the defining set of equations, with one restriction: only terms consisting of 0, the successor S and variables can be used in the inference rules dealing with quantifiers, namely universal elimination and existential introduction. We call such terms basic. Provably recursive functions is a classic topic in proof theory [1]. Let T (e,~x,y) be an arithmetic formula expressing that a deterministic Turing machine with a code e terminates on inputs ~x producing a computation trace with code y. A function f is a provably recursive function of an arithmetic theory T if T ` ∀~x∃y T (e,~x,y) (1)
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