| Omega-consistent Theory |
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In Mathematical Logic , an omega-consistent (or '''ω-consistent''') '''theory''' is a Theory (collection of Sentences ) that is not only Consistent (that is, does not prove a Contradiction ), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. Specifically, if ''T'' is a theory that interprets arithmetic (that is, there is a way to understand some of its objects of discourse as Natural Number s), then ''T'' is omega-inconsistent if, for some property ''P'' of natural numbers (definable in the language of ''T''), ''T'' proves ''P''(0), ''P''(1), ''P''(2), and so on (that is, for every natural number ''n'', ''T'' proves that ''P''(''n'') holds), but ''T'' also proves that there is some natural number ''n'' such that ''P''(''n'') ''fails''. This may not lead directly to an outright contradiction, because ''T'' may not be able to prove for any ''specific'' value of ''n'' that ''P''(''n'') fails, only that there ''is'' such an ''n''. ''T'' is omega-consistent if it is ''not'' omega-inconsistent. EXAMPLE Write PA for the theory Peano Arithmetic , and Con(PA) for the statement of arithmetic that formalizes the claim "PA is consistent". Con(PA) could be of the form "For every natural number ''n'', ''n'' is not the Gödel Number of a proof from PA that 0=1". (This formulation uses 0=1 instead of a direct contradiction; that gives the same result, because PA certainly proves ¬0=1, so if it proved 0=1 as well we would have a contradiction, and on the other hand, if PA proves a contradiction, then it proves anything, including 0=1.) Now, assuming PA is really consistent, it follows that PA+¬Con(PA) is also consistent, for if it were not, then PA would prove Con(PA), contradicting Gödel's Second Incompleteness Theorem . However, PA+¬Con(PA) is ''not'' omega-consistent. This is because, for any particular natural number ''n'', PA+¬Con(PA) proves that ''n'' is not the Gödel number of a proof that 0=1 (PA itself proves that fact; the extra assumption ¬Con(PA) is not needed). However, PA+¬Con(PA) proves that, for ''some'' natural number ''n'', ''n'' ''is'' the Gödel number of such a proof (this is just a direct restatement of the claim ¬Con(PA) ). |
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