Is the axiom of the solvability of every problem a peculiar characteristic of mathematical thought alone, or is it possibly a general law inherent in the nature of the mind, that all questions which it asks must be answerable?
In line with that presumption, Godel was not content merely to demonstrate the possibility of alternative models - to establish, that is, the consistency of the underlying theories; he sought to determine which model was the correct one. His consummate faith (so often vindicated) in the power of his own mathematical intuition led him to search for axioms from which the Continuum Hypothesis might be decided, to find a consistency proof for arithmetic based on constructively evident, though abstract, principles...What is most striking about Godel's philosophical stance is his resolute optimism in confronting the implications of his own results. Having demonstrated that the axiomatic method is fundamentally inadequate for number theory insofar as the truths of arithmetic cannot all be obtained as theorems within any fixed recursively axiomatized system, he saw no reason for despair. In contrast to Post and Turing, he construed the incompleteness theorems not as establishing limitations on the power of human reasoning, but rather as showing "that the kind of reasoning necessary in mathematics cannot be completely mechanized" and thus as affirming the role of the human intellect in mathematical research. Because of his belief that our minds are "not static, but constantly developing," he was confident that new mathematical insights [the "missing axioms"] - principles that "force themselves upon us as being true" - would continue to emerge.
-John Dawson, p.263
