By Torkel Franzén
One of the expositions of Gödel's incompleteness theorems written for non-specialists, this e-book stands aside. With remarkable readability, Franzén offers cautious, non-technical motives either one of what these theorems say and, extra importantly, what they don't. No different booklet 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 numerous spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it's a important addition to the literature. --- John W. Dawson, writer of Logical Dilemmas: The existence and paintings of Kurt Gödel
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Additional resources for Gödel's theorem : an incomplete guide to its use and abuse
That this problem is unsolvable in ZFC was proved using set-theoretical methods introduced by G¨ odel in 1938 and Paul Cohen in 1963, not with the help of the incompleteness theorem. Since ZFC is known to encompass all of the methods of “ordinary” mathematics, what this means is that in order to prove or disprove the continuum hypothesis, new mathematical axioms or principles of reasoning must be introduced. Since it is not part of ordinary mathematical activity to ﬁnd and propose such axioms or principles, it is understandable that mathematicians in general tend to regard a problem that is known to be unsolvable in ZFC as no longer posing a mathematical problem.
6. The Second Incompleteness Theorem 37 Proving ConS That a formal system S is inconsistent means that there are two proofs in the system such that one proves A and the other proves not-A, for some sentence A. Since the property of being the G¨ odel number of a proof in S is required to be a computable one, it follows that “S is consistent” can be formulated as a Goldbach-like statement: it is not the case that there are numbers n and m such that n is the G¨ odel number of a proof in S of A and m is the G¨ odel number of a proof in S of not-A, for the same statement A.
Very often in discussions of the incompleteness theorem it is regarded as unclear what might be meant by saying that an arithmetical statement which is undecidable, say in PA, is true. What, for example, are we to make of the reﬂection that the twin prime conjecture may be true, but undecidable in PA? In saying that the twin prime conjecture may be true, do we mean that it may be provable in some other theory, and if so which one? Do we mean that we may be able to somehow “perceive” the truth of the twin prime conjecture, without a formal proof?