Abstract
Publisher Summary The method of formalizing mathematical proofs gives rise to various formal systems in which certain methods of derivation are delimited in a natural way. Within some of these systems, the chapter is able to formalize all demonstrations of classical number theory, analysis, and set theory. From Godel's incompleteness theorems, it is know that there is no consistent formal system that allows to derive every true arithmetic formula. This can be expressed also by saying that the set of Godel numbers is not recursively enumerable. In a less technical formulation, the number-theoretic concept of truth cannot be operatively defined—that is, the true arithmetic propositions cannot be characterized as those obtained from some explicitly stated propositions and propositional schemata by some rules of procedure given in advance. A complete system (a system that cannot consistently be extended by adding an axiom, which is expressible as a formula of the system) can admit different nonisomorphic models.
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