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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Woodin cardinals

Small forcing creates neither strong nor Woodin cardinals

Posted on September 25, 2011 by Joel David Hamkins
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[bibtex key=HamkinsWoodin2000:SmallForcing]

After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals.

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Posted in Publications | Tagged forcing, indestructibility, W. Hugh Woodin, Woodin cardinals | Leave a reply

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  • Comment by Joel David Hamkins on Transcendence degree of the surreals over the subfield generated by the ordinals
    I would appreciate fuller explanation of this answer, if it would be possible.
  • Comment by Joel David Hamkins on Are there substantive differences between the different approaches to "size issues" in category theory?
    @JoeLamond You say it is reflexive if $\kappa$ is small, but the link you provided said it is reflexive only if $\kappa$ is small. Those are not the same, so I am not sure what the true state is. But also, neither of those statements state whether there is a universal claim to be made. […]
  • Comment by Joel David Hamkins on Forcing with strong binary trees
    Can one implement the fusion arguments? It seems delicate to enforce the strong splitting requirement...
  • Comment by Joel David Hamkins on Interpretability and relative consistency with Kolmogorov randomness axioms
    Can you tell us what is $R$?
  • Comment by Joel David Hamkins on What tools are there, apart from using the countable chain condition, to show that forcing preserves cardinals?
    Perhaps someone should collect the various ideas in the comments and post an answer? I think the site works better when answers are posted as answers.
  • Comment by Joel David Hamkins on What tools are there, apart from using the countable chain condition, to show that forcing preserves cardinals?
    In case you are not aware, the generalized Delta-system lemma (theorem 9.19 in Jech) is extremely useful for proving instances of $\delta$-c.c. for higher cardinals, including your case. Also, an often useful variation of closure would be strategic closure, which shows that no new sequences of a certain length over the ground model are added, […]
  • Comment by Joel David Hamkins on Full name (in the sense of forcing) for a partial order
    This is also possible, since we can mix the names $\tau$ that I had used, with condition $p$, with the name $1_\pi$ having value $\neg p$. That was my first idea, actually, but I realized this complication was not needed for the version of fullness you had stated.
  • Answer by Joel David Hamkins for Full name (in the sense of forcing) for a partial order
    Suppose that we have a $\newcommand\P{\mathbb{P}}\P$-name of a partial order $\langle\pi,\leq_\pi,1_\pi\rangle$. So it is forced that this is a partial order and also that $1_\pi\in\pi$. Let $\pi'$ consist of all nice names $\tau$ for elements of $\pi$. These are the names $\tau$ for which there is a condition $p$ and maximal antichain of conditions $r\leq […]

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