Gödel showed in 1930 that these axioms of identity are complete. That is, every valid schema or sentence of the logic of identity can be proved from them by the logic of quantification(mol273)
Gödel showed in 1930 that these axioms of identity are complete. That is, every valid schema or sentence of the logic of identity can be proved from them by the logic of quantification(mol273)
Godel's proof of the impossibility of a complete consistent systematization of arithmetic depended on constructing a modal of concatenation theory within arithmetic(slp70)
Few of us would have been surprised to learn that there is no algorithm-no effective criterion, no outright test- for truth in elementary number theory. Such a test would make short work of unsolved problems such as Goldbach's conjecture and Fermat's Last Theorem; too good to be true. On the other hand, Godel's theorem came as a shock, for we supposed that truth in mathematics consisted in demonstrability. Substantially the same argument that establishes Godel's theorem, however, establishes something stronger: that truth for elementary number theory cannot be defined in protosyntax at all, either by proof procedure or otherwise(slp236)
Godel showed, on the basis of the Zermelo-Fraenkel set theory plus his axiom of constructibility how the whole universe of sets could be well-ordered. This means that among any lot of sets there is a first, with respect to that order. There are no infinite descents. Moreover, this firstness can be expressed in the set-theoretic notation(slp248)
Godel adduced a theorem that a mathematical theory cannot ordinarily be proved to be free of internal contradiction except by resorting to another theory that rests on stronger assumptions, and hence is less reliable, than the theory whose consistency is being proved. Like the incompletability theorem, this corollary has a melancholy ring. Still it has been found to be of positive utility when we are concerned to prove that one theory is stronger than another: we can do so by proving in the one theory that the other is consistent(tt144)