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  • Is there a set of axioms possible in which CH and GCH are provable (and we can go on with our lives) ?
  • If so, why don't we use this set of axioms ? (i.e., what goes "wrong" with the rest of mathematics).
  • What can be said about large cardinals in a set of axioms under which CH / GCH are provable ?
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  • $\begingroup$ Try to subdivide your question in several questions for better answers. $\endgroup$ – Josué Tonelli-Cueto Mar 9 '14 at 15:56
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See this answer for reasons why some set theorists favor $\mathsf{CH}$ and some reject it. This should give you an idea why it is not simply a matter of adding an axiom that settles the question.

Anyway, yes, there are statements that imply $\mathsf{GCH}$. One is to state that $V=L$ or, more generally, that $V$ is a core model. Woodin has suggested a version of this statement, $V=$Ultimate $L$, that is also expected to be compatible with all known large cardinals. See here for some information, but the statement is technical.

Large cardinals are expected not to have a direct influence on the continuum problem, and also we expect additional axioms to be consistent with arbitrarily strong large cardinals if we are to consider them likely candidates to add to the usual list. Foreman, Magidor, Shelah proposed years ago a "maximality axiom" that implies that $\mathsf{GCH}$ fails everywhere and that there are no inaccessibles. But the axiom was not suggested as a serious candidate, but more as an interesting statement, whose consistency remains in question.

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Yes. $\sf ZFC+GCH$ can easily prove $\sf GCH,CH$ and so on. More to the point, Godel's axiom $V=L$ implies $\sf GCH$ and much more.

We don't often use that because there is a very interest world in set theory beyond just proving $\sf GCH$. In a lot of sense, we care about unprovability almost as much (and sometimes more than) as we care about provability. Axioms like $V=L$ decide many things in set theory, but they also limit things. For example, $V=L$ is incompatible with many large cardinal notions (although $\sf GCH$ is generally compatible).

There are things like forcing axioms which can prove $2^{\aleph_0}=\aleph_2$, and thus disprove $\sf GCH$ and $\sf CH$. These are things we are interested in as well.

Finally, large cardinals are objects, or axioms, whose consistency strength exceed that of $\sf ZFC$. But $\sf GCH$ does not increase the consistency strength, and therefore we cannot prove much more than before (with the slight exception that "weakly inaccessible" are "strongly inaccessible" and similar notions).

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  • $\begingroup$ That last paragraph is problematic, I think. Consistency strength should not be the only guiding force when searching for new axioms. Explanatory power seems far more important. Also, what precisely do you mean by "we cannot prove much more than before"? We can prove much that is not provable in $\mathsf{ZFC}$, much that is of direct relevance to working mathematicians. $\endgroup$ – Andrés E. Caicedo Mar 9 '14 at 16:10
  • $\begingroup$ Yes, you're right. But I have to go and teach soon. I'll revise this after that. $\endgroup$ – Asaf Karagila Mar 9 '14 at 16:14

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