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 On 01/08/17 4:39 AM, Patrik Eklund wrote:
Since the Incompleteness Theorem uses the Liar Paradox, why is it called the Incompleteness Theorem and not the Incompleteness Paradox?
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