On the Original Gentzen Consistency Proof for Number Theory

Abstract

Publisher Summary This chapter investigates the original Gentzen consistency proof for number theory. Gentzen consistency proof for the formal system of first order number theory, including standard logic, the Peano axioms and recursive definitions is considered. The chapter describes Gentzen's original proof. Gentzen admits arithmetical function symbols (with the restriction that for numerical arguments the value of the function must be computable), thus, there is no loss of generality in assuming that all prime formulas are equations between terms. A reduction process for a sequent is a procedure consisting of a terminating sequence of successive reduction steps by which the sequent is brought into a final form (Endform). It means to a sequent satisfying at least one of the two conditions: (1) that the succedent is a true numerical equation or (2) that some antecedent formula is a false numerical equation.