Sorry if this is a double-post, but it appears a version I attempted to send earlier never left my machine.
Am Jan 24, 2012 um 9:34 PM schrieb Rich Cooper:
Gödel showed...
Seriously, Rich?
...that any logical system at least as
powerful as arithmetic is necessarily conflicted;
there are true theorems that cannot be proven and
there are false theorems that cannot be refuted.
He showed no such things:
1. "any logical system at least as powerful as arithmetic"
No. All that is necessary for incompleteness is a very small, finite fragment of arithmetic, often known as "Robinson Arithmetic" or "Q". "Arithmetic" per se, as usually defined, is precisely what Gödel showed cannot be captured fully in an axiomatic system, namely, the set of all truths about the natural numbers expressed in the language containing the numeral 0, the successor operator, and the symbols for addition and multiplication.
2. "any logical system at least as powerful as arithmetic is necessarily conflicted"
"Conflicted" is not a meaningful mathematical notion. And insofar as it is meant to be an impressionistic or evocative description of an incomplete system, it is wildly inappropriate. "Conflict" suggests some sort of contradiction or paradox. Nothing of the sort arises in incomplete systems. Indeed, quite the opposite: Incompleteness implies consistency.
3. "there are true theorems that cannot be proven"
This is, alas, incoherent. It makes no sense to say that a statement is a theorem (let alone a "true" theorem), full stop. A statement can only be a theorem relative to some system; the theorems of the system are, by definition, the statements that can be proved in the system. So it is, by definition, impossible for a theorem (of some system) to be unprovable (in that system) -- though, of course, a theorem of one system might be unprovable in *another* system.
4. "there are false theorems that cannot be refuted"
See previous.
Here is (a still somewhat informal version of) the actual theorem, where a "system" is an axiomatic theory (with a decidable set of axioms) built on first-order logic:
(GT) For any consistent system S containing at least Robinson Arithmetic, there are sentences in the language of S that S neither proves nor refutes.
And from (GT), something vaguely like your statement 3 above follows as a corollary:
(GTC) For any consistent system S containing at least Robinson Arithmetic, there are sentences in the language of S that are true (in the natural numbers) but which are not theorems of S.
But incompleteness is not the same as ambiguity.
Neither is it the same as acceleration, electricity, or good health, to all of which it is equally (ir)relevant.
In effect, Gödel showed that, given a single
observer (supposedly objective and universal in
her language mappings and trained in mathematical
logic), even the single observer has an incomplete
grasp of proofs based on FOL+arithmetic.
No, he showed absolutely no such thing, in effect or otherwise. Gödel's incompleteness theorem has *absolutely nothing whatever* to do with observers and their graspings of proofs, straws, or their own bootstraps. Gödel's theorem is a mathematical theorem about certain types of mathematical objects, viz., axiomatic systems. It has no more to do with observers than does the proof that the square root of 2 is irrational or that there are infinitely many prime numbers.
I have no interest in continuing this discussion; interested readers can dig through the archives to peruse the thread from a year or so ago when this came up then and see how all of that played out. But I DO have a sincere recommendation for you, namely, the marvelous little book Gödel's Theorem: An Incomplete Guide to Its Use and Abuse by the brilliant and sorely missed Swedish logician Torkel Franzen. It is not only perhaps the best "popular" exposition of Gödel's theorem ever written, it includes a comprehensive overview of the manifold ways in which the theorem has been misunderstood, misinterpreted, and (often hilariously) exploited for quasi-philosophical gain. From the introduction:
"[A]mong the nonmathematical arguments, ideas, and reflections inspired by Gödel's theorem there are also many that...occur naturally to many people with very different backgrounds when they think about the theorem. Examples of such reflections are 'there are truths that logic and mathematics are powerless to prove,' 'nothing can be known for sure,' and 'the human mind can do things that computers can not.' The aim of the present addition to the literature on Gödel's theorem is to set out the content, scope, and limits of the incompleteness theorem in such a way as to allow a reader with no knowledge of formal logic to form a sober and soundly based opinion of these various arguments and reflections invoking the theorem. To this end, a number of such commonly occurring arguments and reflections will be presented, in an attempt to counteract common misconceptions and clarify the philosophical issues."
Very highly recommended.
-chris |
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