Talk:Gödel's incompleteness theorems/Arguments

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This page is for arguments over the validity of Gödel's incompleteness theorems. This is not an archive; you may feel free to edit this page. Please use this page for comments not directly relevant to improving the article Gödel's incompleteness theorems.

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[edit] is the incompleteness theorem complete?

This is not an argument against the validity of the incompleteness theorem, and therefore it does not belong in the arguments section. This is a question about the incompleteness theorem and what it means in its own terms. What the theorem says about itself in its own terms is important to the discussion of the article.

The question is, according to the terms of the incompleteness theorem, does the incompleteness theorem apply to itself, and if so, does that make the theorem itself complete, or incomplete? Secondly, if the incompleteness theorem is itself complete, does that make it consistent, or inconsistent? Lastly, how do we really know? I would appreciate it if somebody could give me a straight answer because it is important to the discussion of the article and what it says about itself in its own terms.

Your question is ill-formed. Completeness and consistency are properties of theories. The incompleteness theorem is not a theory, but a proposition. --Trovatore 05:00, 29 January 2007 (UTC)

You may of course consider the incompleteness theorem (for a given theory T, say PA or PM or ZFC etc) as a theory T' with a single axiom. (Not that that would make much sense...)

That is, T' = { A } , where A is the sentence saying that T is incomplete.

For the example theories T mentioned above, T' is weaker than "T + Consistent(T)", and is therefore also an incomplete theory.

(However, depending on the exact formulation of the sentence A, T' may not include enough arithmetic, and may therefore not be essentially incomplete.)

--62.233.163.82 19:44, 10 February 2007 (UTC)

Dear Trovatore, You must explain the difference between a theory and a proposition. —The preceding unsigned comment was added by 72.16.111.54 (talk • contribs) 18:56, 23 March 2007 (UTC)

Sure thing. A theory is a collection of propositions. --Trovatore 19:05, 23 March 2007 (UTC)

[edit] I don't get it, this seems simple to fix

In these terms, the Gödel sentence is a claim that there does not exist a natural number with a certain property. A number with that property would be a proof of inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to hypothesis. So there is no such number, and the Gödel statement is true, but the theory cannot prove it.

Ok, so let's say, "There does not exist a number that is prime, higher than 2 and is divisible by 2."

IF we found such a number, then the statement is false. However, since we can't "prove a negative", thus we can't ever *prove* this to be true. Tell me if I am sorta right so far (this might be a retarded example, I'm not a mathematician, just trying to muddle through the logic)?

SO, why can't we just prove the exact reverse of this? i.e. Given an infinite amount of time, we look at ALL the prime numbers greater than 2 and show that all of them are not divisible by 2.

It seems to my logic that when you've got infinity to play with, showing that *all* possible numbers do not have the characteristic is the same as saying that a possible number *cannot* have that characteristic. —The preceding unsigned comment was added by 66.159.227.60 (talk • contribs) 19:36, 1 April 2007 (UTC)

The first big problem in the above is the canard that you can't "prove a negative". In mathematics we prove negative statements all the time, and proving that there is no number with the property you give is in fact very easy. --Trovatore 23:21, 1 April 2007 (UTC)

Yes, that example seems much too simple. I'm not a mathematician either thus I don't understand the theory completely. However to get around the infinity statement, you could just change it to "There does not exist a prime number, higher than 2, less than 100, and is divisible by 2." But I doubt that this is what the inconsistency theory is talking about.Niubrad 23:59, 1 April 2007 (UTC)