Axiom of Reducibility - Criticism of The Axiom of Reducibility - David Hilbert 1927

David Hilbert 1927

David Hilbert's axiomatic system that he presents in his 1925 The Foundations of Mathematics is the mature expression of a task he set about in the early 1900s but let lapse for a while (cf his 1904 On the foundations of logic and arithmetic). His system is neither set theoretic nor derived directly from Russell and Whitehead. Rather, it invokes 13 axioms of logic—four axioms of Implication, six axioms of logical AND and logical OR, 2 axioms of logical negation, and 1 ε-axiom ("existence" axiom)-- plus a version of the Peano axioms in 4 axioms including mathematical induction, some definitions that "have the character of axioms, and certain recursion axioms that result from a general recursion schema" plus some formation rules that "govern the use of the axioms".

Hilbert states that, with regard to this system, i.e. "Russell and Whitehead's theory of foundations ... the foundation that it provides for mathematics rests, first, upon the axiom of infinity and, then upon what is called the axiom of reducibility, and both of these axioms are genuine contentual assumptions that are not supported by a consistency proof; they are assumptions whose validity in fact remains dubious and that, in any case, my theory does not require . . . reducibility is not presupposed in my theory . . . the execution of the reduction would be required only in case a proof of a contradiction were given, and then, according to my proof theory, this reduction would always be bound to succeed."

It is upon this foundation that modern recursion theory rests.