Monday, February 2, 2009

Chew on this…

Gödel's second incompleteness theorem says that a recursively axiomatizable system that can interpret Robinson arithmetic can prove its own consistency only if it is inconsistent. Moreover, Robinson arithmetic can be interpreted in general set theory, a small fragment of [Zermelo-Fraenkel theorem with the axiom of choice]. Hence the consistency of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent). Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics. The consistency of ZFC does follow from the existence of a weakly inaccessible cardinal, which is unprovable in ZFC if ZFC is consistent. Nevertheless, it is unlikely that ZFC harbors an unsuspected contradiction; if ZFC were inconsistent, it is widely believed that that fact would have been uncovered by now. This much is certain—ZFC is immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox.

Robinson arithmetic, or Q, is a finitely axiomatized fragment of Peano arithmetic (PA), first set out in Robinson (1950). Q is essentially PA without the axiom schema of induction.

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. … The Peano axioms contain three types of statements. The first four statements are general statements about equality; in modern treatments these are often considered axioms of pure logic. The next four axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic [PA] is obtained by replacing this second-order induction axiom with a first-order axiom schema.

All quotations cribbed from Wikipedia.

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