Talk:List of statements undecidable in ZFC

Remarks on assumptions used to prove independence

I've revised the paragraph on large cardinals. What it said before was true--ZFC can neither prove nor disprove the existence of large cardinals. However what the page said at the top was "assuming ZFC is consistent", which in context probably refers to statements that can be formally proved in ZFC+Con(ZFC), and it is not possible to prove in ZFC+Con(ZFC) that the existence of large cardinals is consistent. In some sense that's the whole point of large cardinals--they provide a scale by which to measure "consistency strength".

But I think maybe the premise of the page should be rethought. It's extremely arbitrary--especially when talking aobut results like these--to assume tacitly that what we can assert is just what's provable in ZFC, and then add "assuming ZFC is consistent" to bump it up to ZFC+Con(ZFC). There are quite a few interesting statements that are independent of ZFC, but to prove one direction or the other you need some large cardinal strength; these could naturally be included here. --Trovatore 16:13, 15 October 2005 (UTC)

I thought it was not currently known whether ZFC disproves the existence of any type of large cardinals, unless this is a remarkable recent development of which I'm not aware. If it were true that ZFC cannot disprove the existence of large cardinals then, assuming ZFC is consistent, there must be a model of ZFC in which there are no large cardinals. I've never seen one; could you point me to an example? (For example I would like to see a model that has no strongly inaccessible cardinals that does not require the existence of strongly inaccessible cardinals to construct.) --Shawn

Here's an example of how to construct one: either there is a strongly inaccessible cardinal or not. If not, then V is a (class) model of ZFC + "there are no strongly inaccessible cardinals." If yes, let \lambda be the least. Then V_\lambda is such a model. This could be strengthened to weakly inaccessibles by working in L. 128.237.246.219 (talk) 00:22, 13 October 2010 (UTC)

Fubini theorem

I've always thought that the consistency of the Fubini theorem for any function was first proved by Harvey Friedman: A Consistent Fubini-Tonelli Theorem for Nonmeasureable Functions, Illinois J. Math., Vol. 24, No. 3, (1980), pp. 390-395.Kope (talk) 07:17, 10 July 2008 (UTC)

Right. I'll add it (but keep the Freiling paper which is quite popular). Uffish (talk) 09:57, 20 July 2008 (UTC)

Exact Godel numbers needed

For all the undecidables mentioned, no exact Godel numbers are given. They should be, with the details of the calculation in each case. —Preceding unsigned comment added by 81.158.135.109 (talk) 09:22, 21 July 2009 (UTC)

• That would be impractical, because even for relatively simple formulae (e.g. 0=0), the corresponding Gödel number is quite large. And what would be the point anyway? The undecidability of these statements hasn't been derived from Gödel's incompleteness theorem. - Mike Rosoft (talk) 12:01, 25 June 2010 (UTC)

Mike Rosoft has used the singular "Godel number". Actually, there are infinitely many Godelisations and the same equation such as 0=0 has infinitely many Godel numbers. —Preceding unsigned comment added by 86.177.254.83 (talk) 11:15, 3 July 2010 (UTC)

Fine, but so what? Regardless of the numbering scheme used, the Goedel numbers are still irrelevant. --Trovatore (talk) 20:37, 3 July 2010 (UTC)