List of Statements Independent of ZFC
en.wikipedia.org
List of Statements Independent of ZFC
1–10 of 108 posts
Re: List of Statements Independent of ZFC
#2Hopefully this will clear at least some of the Gödelian misconceptions that pop up from time to time here
Re: List of Statements Independent of ZFC
#3A very interesting discussion about this topic that also includes many examples and references is available on MathOverflow:
What are some reasonable-sounding statements that are independent of ZFC?
https://mathoverflow.net/questions/1924/what-are-some-reason...
With the top voted result currently being:
"If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y."
As is also mentioned in the linked entry, this statement is independent of ZFC.
Re: List of Statements Independent of ZFC
#4Some of this was familiar, but I also saw this:
> In 1973, Saharon Shelah showed that the Whitehead problem ("is every abelian group A with Ext^1(A, Z) = 0 a free abelian group?") is independent of ZFC.
(This is the same Shelah who proved what is usually called "Sauer's Lemma" on set separability, upon which the "VC dimension" and the resulting VC learning theory are based.)
Anyway, this was surprising because I didn't know there were any theorems outside of set theory that had been known and studied, and then later turned out to be independent of ZFC.
Apparently, the surprise I felt is just because I wasn't paying attention -- there are a couple of other problems in the page that seem to have a similar flavor. I'm not enough of a mathematician to appreciate to what extent they are interesting problems that arise independently of counterexamples.
[edited to add: the comment of @triska nearby addresses exactly this point!]
Re: List of Statements Independent of ZFC
#5A very interesting discussion about this topic that also includes many examples and references is available on MathOverflow: What are some reasonable-sounding statements that are independent of ZFC? https://mathoverflow.net/questions/1924/what-are-some-reason... With the top voted result currently being: "If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y." As is also mentioned i…
Cantor bad.
Re: List of Statements Independent of ZFC
#6A very interesting discussion about this topic that also includes many examples and references is available on MathOverflow: What are some reasonable-sounding statements that are independent of ZFC? https://mathoverflow.net/questions/1924/what-are-some-reason... With the top voted result currently being: "If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y." As is also mentioned i…
That is a strange one.
Re: List of Statements Independent of ZFC
#7A very interesting discussion about this topic that also includes many examples and references is available on MathOverflow: What are some reasonable-sounding statements that are independent of ZFC? https://mathoverflow.net/questions/1924/what-are-some-reason... With the top voted result currently being: "If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y." As is also mentioned i…
Cantor bad.
What does this have to do with Cantor?