The consistency of inconsistency claims

A theory is inconsistent if we can prove a contradiction using basic logic and the principles of that theory. Consistency is a much weaker condition that truth: if a theory T is true, then T
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A theory is inconsistent if we can prove a contradiction using basic logic and the principles of that theory. Consistency is a much weaker condition that truth: if a theory T is true, then T consistent, since a true theory only allows us to prove true claims, and contradictions are not true. There are, however, infinitely many different consistent theories that we can construct using, for example, the language of basic arithmetic, and many of these are false. That is, they do not accurately describe the world, but are consistent nonetheless (one way of understanding such theories is that they truly describe some structure similar to, but distinct from, the standard natural number structure). In 1931 Kurt Gödel published one of the most important and most celebrated results in 20th century mathematics: the incompleteness of arithmetic. Gödel’s work, however, actually contains two distinct incompleteness theorems. The first can be stated a bit loosely as follows: First Incompleteness. . .

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News source: OUPblog » Philosophy

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