Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
  
  
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeDec 31st 2011
   • (edited Dec 31st 2011)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI don't know what to make of this, added to [[regular cardinal]] by David Roberts: >Assuming the consistency of 'there is a [[proper class]] of [[strongly compact cardinal|strongly compact cardinals]]', it is consistent that every uncountable cardinal is singular, a result due to [[Moti Gitik]]. How does this square with the easy proof that $\aleph_1$ is uncountable and regular? (And of course $\aleph_\omega$ is uncountable and singular, so it's not a matter of getting it exactly backwards.) So in what context is this occurring, or what is it supposed to say?

   I don’t know what to make of this, added to regular cardinal by David Roberts:

   Assuming the consistency of ’there is a proper class of strongly compact cardinals’, it is consistent that every uncountable cardinal is singular, a result due to Moti Gitik.

   How does this square with the easy proof that is uncountable and regular? (And of course is uncountable and singular, so it’s not a matter of getting it exactly backwards.) So in what context is this occurring, or what is it supposed to say?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeDec 31st 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI'm guessing that Gitik is working in the absence of AC?

   I’m guessing that Gitik is working in the absence of AC?

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeJan 1st 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexThat's certainly one possibility, but if so, then the phrasing is quite odd. In any case, we must ask, consistent *with what*?

   That’s certainly one possibility, but if so, then the phrasing is quite odd. In any case, we must ask, consistent with what?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJan 1st 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexMy guess would be, consistent with ZF. But David should tell us, since he presumably looked at the paper. (-:

   My guess would be, consistent with ZF. But David should tell us, since he presumably looked at the paper. (-:

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeJan 1st 2012
   • (edited Jan 1st 2012)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexHe also might have read [Gitik's bio on Wikipedia](https://en.wikipedia.org/wiki/Moti_Gitik), which is equally vague. At least that makes it unlikely to be a mistake, so it probably is just all over $ZF$, as you suggested.

   He also might have read Gitik’s bio on Wikipedia, which is equally vague. At least that makes it unlikely to be a mistake, so it probably is just all over , as you suggested.

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 1st 2012
   • (edited Jan 2nd 2012)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexSorry for the vagueness. The correct statement is: > Con(ZFC + there is an unbounded class of strongly compact cardinals) $\Rightarrow$ Con(ZF + $Cof(\aleph_\alpha) = \aleph_0$ for all ordinals $\alpha$) and hence the only regular cardinal in this model is $\aleph_0$. Got to run, I can edit the page later if needed.

   Sorry for the vagueness. The correct statement is:

   Con(ZFC + there is an unbounded class of strongly compact cardinals) Con(ZF + for all ordinals )

   and hence the only regular cardinal in this model is . Got to run, I can edit the page later if needed.

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeJan 2nd 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexThanks, David; I've edited the page.

   Thanks, David; I’ve edited the page.