Talk:List of statements independent of ZFC

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Remarks on assumptions used to prove independence[edit]

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)[reply]

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)[reply]

@Trovatore: you said above "ZFC can neither prove nor disprove the existence of large cardinals". Surely, this can only be true if ZFC is consistent; otherwise ZFC can prove and disprove anything, including its own consistency. So to me the addition in the intro ("assuming ZFC is consistent") makes perfect sense. If instead we made a list of "statements which are known to be neither provable nor disprovable in ZFC", then the list would be empty. BTW, with Shawn above, I believe that currently it is not known whether ZFC+Con(ZFC) can disprove the existence of large cardinals. AxelBoldt (talk) 00:21, 20 January 2015 (UTC)[reply]

OK, here's the situation: ZFC cannot prove the existence of large cardinals unless ZFC is inconsistent (which, of course, it is not). It cannot disprove it either, unless of course their existence is inconsistent (which, of course, it is not).

You might wonder if something like the following could be true: You can't prove from ZFC that there is a (say) Mahlo cardinal, but you can prove that it's consistent that there's a Mahlo cardinal. The answer to that is no. There is no hope to prove any consistency result like that; it contradicts second incompleteness, because Con(ZFC+Mahlo) is actually consistency-wise slightly stronger than ZFC+Mahlo.

I object somewhat to the "not known" formulation. It depends on what you mean by "known". There is a tradition that equates "known" with "proved", but to make that work you have to specify what axioms you're considering to be "known" in the first place, so this is not really a justifiable epistemology.

I hope you agree that the edit I made makes this clearer. --Trovatore (talk) 04:00, 25 January 2015 (UTC)[reply]

@Trovatore: Sure, your edit is fine and I agree with everything you said. The point that a consistent ZFC can never be hoped to prove the relative consistency of large cardinals is an important one; maybe we should add it here or at large cardinal. Cheers, AxelBoldt (talk) 00:29, 27 January 2015 (UTC)[reply]

Fubini theorem[edit]

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)[reply]

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

Exact Godel numbers needed[edit]

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)[reply]

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)[reply]

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)[reply]

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

MA and CH[edit]

I think the current text about the chain of implications, with Martin's axiom at the end, is a little — peculiar. It's all true, of course, formally. But the implication CH->MA is true for an uninteresting reason. CH essentially makes MA vacuous.

In actual practice, consequences of MA are interesting only if CH fails. The weakest form of MA that is used in practice is MA(ℵ₁), which implies (for example) that all sets of reals of size ℵ₁ have measure 0 (and thus obviously contradicts CH).

The diamond-shaped graph is sort of interesting, I guess — V=L implying two different strengthenings of CH, neither of which implies the other — but I don't see the point of putting MA at the end of it. --Trovatore (talk) 20:15, 3 May 2017 (UTC)[reply]

Undecidable diophantine equation[edit]

"One can write down a concrete polynomial p ∈ Z[x₁, ..., x₉] such that the statement "there are integers m₁, ..., m₉ with p(m₁, ..., m₉) = 0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent). The polynomial is constructed so that it has an integer root if and only if ZFC is inconsistent."

Sorry but what does this mean? Since we are assuming ZFC is consistent, ZFC proves that p(m₁, ..., m₉) = 0 must have no solution by the second sentence! After all, how is the set of natural numbers defined if ZFC is inconsistent? 129.104.241.198 (talk) 03:50, 12 September 2023 (UTC)[reply]