One (rather trivial) reason why Goedel's Theorem isn't a paradox is because it's true, and there's a good argument for its truth, indeed one that can be formalised, but not in the system that the theorem is about. This differentiates it from paradoxes such as that of the liar, which can't consistently be assigned any truth value. However, there's something deeper here, which is that most paradoxes go against common sense, so that discovering them is a sort of self-limitation of reason (showing that something which seemed obvious is in fact false). And so was Goedel's theorem: nobody in the Hilbert school seems to have thought seriously that their program could fail, just that they hadn't quite got it right yet. And it says a great deal for Hilbert's personal qualities that he accepted Goedel's theorem when he saw it. (See Wilfried Sieg's wonderful book Hilbert's Programs and Beyond; a must read, in my opinion.) Graham On 08/01/17 23:30, Vaughan Pratt wrote:
What's in a name?
A theorem is only paradoxical when it proves the inconsistency of an otherwise plausible axiom system, for example one that assumes there is a set of all sets with a well-defined cardinality, or Hilbert's conviction that all formally definable problems are solvable, or the "self-evident fact" that a dense linear order has no room to interpolate another number, or that an open cover of the rationals must cover the whole real line (as Pure Maths honours students in 1965 Henry Irgang and I visited Max Kelly in his office after class to express our incredulity), etc. etc.
Any of these could have been officially called the such-and-such paradox. Furthermore only some of them involve the Liar Paradox.
(I taught a freshman seminar titled "Paradox: Bug or Feature?" many years ago loosely based on Mark Sainsbury's book "Paradoxes". A basic example of "Feature" is recursion, associated with the non-existence of a largest integer. True to form I digressed with topics like surreal numbers and other topics the class expressed interest in. The hardest paradox we encountered was the Surprise Exam paradox where the teacher says there will be an exam next week and it will be a surprise---to the pupils' surprise it was given on Tuesday, contrary to a reasonable-looking argument. None of us succeeded in analyzing it as satisfactorily as the other paradoxes we treated. Sol Feferman taught a similar seminar a couple of years later.)
Vaughan Pratt
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]