Talk:Paul Cohen (mathematician)
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"This result is possibly the most famous non-trivial example illustrating Gödel's incompleteness theorem. "
This claim is false. Godel's theorem was about the incompleteness of arithmetic. Cohen's result is that the much stronger theory ZF cannot proved and that ZFC cannot prove the continum hypothesis. But neither Choice nor CH are formulable in arithemtic. Godel's Incompleteness result showed that there are purely arithemtical claims that cannot be decided by arithmetic. The undecidability of CH by ZFC is no more an example of that than is the fact that a theory of arithmetic cannot prove that pi is transcendental.
I will leave it for a week and if no discussion has been generated by then, I will delete it.
It is possible that I went overboard by putting the factual errors boiler plate up. If so, sorry, I am new and learning the ways of the wiki.
vanden 03:51, 4 Nov 2003 (UTC)
- I would suggest the claim is correct. I would state Gödel's incompleteness theorem, in this context, as "Given any sufficiently complex (consistent) axiom set, one can formulate a statement that cannot be either proved or disproved". In particular "There is a statement in ZFC that cannot be either proved or disproved" and the continuum hypothesis is such a statement, thus demonstrating the correctness of Gödel's theorem.
- Note that I wouldn't particularly hate to see the statement deleted, it's just not as inaccurate as you make it sound.
- Prumpf 15:45, 22 Nov 2003 (UTC)
It is surely better to say that this is a celebrated independence result. Then a discussion of independence results can say carefully what the Goedel theorem does and doesn't imply.
Charles Matthews 09:07, 4 Nov 2003 (UTC)
Vanden hasn't contributed since November. His claim on independence of CH not illustrating Goedel seems to me over the top. Certainly Goedel's result is usually thought of as applying to systems 'at least as strong as' arithmetic; so saying it doesn't apply to set theory is making an odd point.
I think the factual accuracy notice here is excessive.
Charles Matthews 09:42, 10 Mar 2004 (UTC)
---
"the most famous non-trivial example"? I don't claim that this is outright false, but it is highly misleading, for the following reasons:
First: Gödel himself gave, for every theory T (that satisfies the assumption of his theorem -- consistent, recursively axiomatized, and encodes arithmetic) a statement that is neither provable nor refutable in T, namely "Con(T)", the formula that (naturally) encodes the consistency of T. (This is the content of his Second Incompleteness Theorem.) The earliest (and arguably the most famous) of those statements is Con(Principia Mathematica), or Con(Peano Arithmetic).
Second: An important point in the incompleteness theorem is the following: not only is there a statement that is neither provable nor refutable in T, but there is a true statement that is not provable (and of course not refutable) in T. In the case of "Con(T)", it is the fact that Con(T) is unprovable (and not that Con(T) is not refutable) that is paradoxical, because we think that T is indeed consistent, i.e., Con(T) is true. -- In the case of CH, however, there is no clear opinion on whether CH is "really" true or not, so it is not clear if it is the nonprovability of CH (proved by Cohen) or the nonprovability of non-CH (proved by Gödel) that shows how limited ZFC's knowledge about "truth" is.
However, Cohen's result was a quantum leap for set theory. He did not get the Fields medal for finding a nice example of incompleteness; he received it for inventing such a powerful new method, and for solving Hilbert's first problem.
Aleph4 21:01, 10 Mar 2004 (UTC)
I would like to comment on the statement - Most famous example proving the incompleteness of a formal system. Godel's theorem assures us of incomplete systems - but that is not the point, but something more. What Godel's theorem tells us is that provability is a weaker notion than truth, ie there are true statements that cannot be proven. In this context, if we knew that for instance CH was true but could not be proven in ZFC, the statement in question would be correct. That is not the case.
User:Mohan Ravichandran 23:30, 18 Feb 2005 (UTC)
Well, either the CH or its negation _is_ "true" -- and neither of them are provable in ZF. So for CH, or for !CH, it is the case.
[edit] Cohen's Quote
In reading the quote by Cohen, I find some words that convey no meaning. To say that a set is "incredibly rich" is to say that any person cannot believe how large a quantity of elements are contained in the set. To say that other people may "express themselves more eloquently" is to say that they will use more appropriate force. But, more force will not result in better proof. Neither of these phrases coveys any clear or intelligible information.64.12.116.74 12:49, 24 April 2006 (UTC)Lestrade
[edit] Deceased?
A recent edit lists Paul Cohen as deceased. A quick check of both google and the Stanford math department home page does not show up with any hits. Is there any source for this info? If not then this may be a hoax. Terry 04:29, 25 March 2007 (UTC)
It s not a hoax. An email was circulated on the Stanford dept mailing list a couple hours before the edit. Obviously the person adding it is well informed.
- An email was also sent to the FOM list with the news of his passing. I am sure that in the coming week there will be an obituary published that we can link to. CMummert · talk 14:06, 25 March 2007 (UTC)
- e-mail at FOM, dated March 24, 2007--Aleph4 12:05, 26 March 2007 (UTC)
