Gödel's theorem : an incomplete guide to its use and abuse by Torkel Franzén

By Torkel Franzén

One of many expositions of Gödel's incompleteness theorems written for non-specialists, this e-book stands aside. With remarkable readability, Franzén offers cautious, non-technical reasons either one of what these theorems say and, extra importantly, what they don't. No different publication goals, as his does, to deal with intimately the misunderstandings and abuses of the incompleteness theorems which are so rife in renowned discussions in their importance. As an antidote to the various spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it's a invaluable addition to the literature. --- John W. Dawson, writer of Logical Dilemmas: The lifestyles and paintings of Kurt Gödel

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The notion of a computable property was introduced earlier in connection with Goldbach-like statements when applied to properties of numbers, but it applies equally to properties of sentences and finite sequences of sentences in a formal language. The concept of computability, and its connection with formal systems and with the incompleteness theorem, will be treated in a more systematic fashion in Chapter 3. 2. The First Incompleteness Theorem 21 S, apart from incorporating a certain amount of arithmetic, is consistent.

Other ways of formulating self-referential sentences using syntactic operations have also been formulated later. For example, the American logician and philosopher W. V. O. ” yields a sentence with property P when appended to its own quotation. The general fixpoint construction is widely used in logic to prove various results. G¨ odel used it to prove his first incompleteness theorem, by applying it to the property of not being a theorem of S. By a G¨ odel sentence for S is meant a sentence G obtained through the general fixpoint construction, such that S proves G if and only if n is not the G¨ odel number of a theorem of S, where n is the G¨ odel number of G itself.

Suppose the consistency of PA were in fact provable in PA itself. Would such a proof have any value as a proof that PA is in fact consistent? Not necessarily, for why should we accept this particular proof in PA? After all, if the consistency of PA is in doubt, the validity of a consistency proof given in PA is equally in doubt—at least as long as we haven’t seen the proof. For there is a loophole: we might consider a consistency proof for PA carried out in PA to be conclusive because it only uses a small part of PA, one that we have no doubt is consistent even if we can muster doubts about the consistency of PA as a whole.

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