Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: naturalist account of forcing

A multiverse perspective in mathematics and set theory: does every mathematical statement have a definite truth value? Shanghai, June 2013

Posted on May 18, 2013 by Joel David Hamkins
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Fudan blueThis will be a talk for specialists in philosophy, mathematics and the philosophy of mathematics, given as part of the workshop Metamathematics and Metaphysics, June 15, 2013, sponsored by the group in Mathematical Logic at Fudan University.

Abstract:  Much of the debate on pluralism in the philosophy of set theory turns on the question of whether every mathematical and set-theoretic assertion has a definite truth value. A traditional Platonist view in set theory, which I call the universe view, holds that there is an absolute background concept of set and a corresponding absolute background set-theoretic universe in which every set-theoretic assertion has a final, definitive truth value. I shall try to tease apart two often-blurred aspects of this perspective, namely, to separate the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique. A competing view, the multiverse view, accepts the former claim and rejects the latter, by holding that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe, and a corresponding pluralism of set-theoretic truths. After framing the dispute, I shall argue that the multiverse position explains our experience with the enormous diversity of set-theoretic possibility, a phenomenon that is one of the central set-theoretic discoveries of the past fifty years and one which challenges the universe view. In particular, I shall argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

Fudan University seal

 

Slides

 

 

 

 

The talk will engage with ideas from some of my recent papers on the topic:

  • The set-theoretic multiverse
  • The multiverse perspective on the axiom of constructibility
  • Is the dream solution of the continuum hypothesis possible to achieve?

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Posted in Talks | Tagged CH, forcing, multiverse, naturalist account of forcing, pluralism, Shanghai | Leave a reply

Well-founded Boolean ultrapowers as large cardinal embeddings

Posted on June 26, 2012 by Joel David Hamkins
2
  • J. D. Hamkins and D. Seabold, “Well-founded Boolean ultrapowers as large cardinal embeddings,” , p. 1–40, 2006.
    [Bibtex]
    @ARTICLE{HamkinsSeabold:BooleanUltrapowers,
    AUTHOR = "Joel David Hamkins and Daniel Seabold",
    TITLE = "Well-founded {Boolean} ultrapowers as large cardinal embeddings",
    JOURNAL = "",
    YEAR = "2006",
    volume = "",
    number = "",
    pages = "1--40",
    month = "",
    note = "",
    eprint = "1206.6075",
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url = {http://jdh.hamkins.org/boolean-ultrapowers/},
    abstract = "",
    keywords = "",
    source = "",
    file = F,
    }

Boolean ultrapowers extend the classical ultrapower construction to work with ultrafilters on any complete Boolean algebra, rather than only on a power set algebra. When they are well-founded, the associated Boolean ultrapower embeddings exhibit a large cardinal nature, and the Boolean ultrapower construction thereby unifies two central themes of set theory—forcing and large cardinals—by revealing them to be two facets of a single underlying construction, the Boolean ultrapower.

The topic of this article was the focus of my tutorial lecture series at the Young Set Theorists Workshop at the Hausdorff Center for Mathematics in Königswinter near Bonn, Germany, March 21-25, 2011.

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Posted in Publications | Tagged Boolean ultrapower, Daniel Seabold, elementary embeddings, forcing, large cardinals, multiverse, naturalist account of forcing | 2 Replies

Recent Comments

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  • Comment by Joel David Hamkins on Which cardinal $\kappa\geq \omega_1$ is critical for the following property...?
    If it is true for $\kappa$, then isn't it also true for any smaller $\kappa$, including $\kappa=\omega_1$? Given any set of size $\omega_1$, first extend it to a set of size $\kappa$, get the $X_\alpha$'s, and then cut back down to the original set. Or have I misunderstood? Oh, maybe when you cut down, you […]
  • Comment by Joel David Hamkins on Statements in differential geometry independent from ZFC
    @BenjaminSteinberg Every computably undecidable decision problem is saturated with logical undecidability, over any base theory, since otherwise we could solve the problem by searching for proofs. In this sense, any computably undecidable problem of differential geometry will provide an answer to the question, even if one uses much stronger theories than ZFC.
  • Comment by Joel David Hamkins on Dedekind-Peano axioms, but numbers have at most one successor
    Ah, that is helpful.
  • Comment by Joel David Hamkins on Dedekind-Peano axioms, but numbers have at most one successor
    One should think of it as a very weak theory, in which even exponentiation is problematic and induction is possible only for very local phenomena.
  • Comment by Joel David Hamkins on Dedekind-Peano axioms, but numbers have at most one successor
    Ah, I thought you were asking about the second order theory. Huge difference. Since the models of the first-order theory are exactly the cut-offs of models of $I\Delta_0$, as I explain in my answer, the question is whether those theorems are provable in $I\Delta_0$, and there are many open questions about that. For example, pigeon-hole […]
  • Comment by Joel David Hamkins on Dedekind-Peano axioms, but numbers have at most one successor
    It is problematic to speak of "provable" in a second-order theory, since we don't have a sound & complete proof system for second-order logic.
  • Comment by Joel David Hamkins on Dedekind-Peano axioms, but numbers have at most one successor
    Both of those will be provable in the second-order theory. For Bertrand, take the statement for every n, there is a prime between n and 2n. You interpret the numbers up to 2n using digit representation as in my answer. This is a valid consequence of PA2top because it is true in standard finite segments. […]
  • Comment by Joel David Hamkins on Simpler proofs using the axiom of choice
    Thanks, Andrej, that is helpful. (I guess you mean the complement in $\mathbb{R}^3$ of the union of the circles?)

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absoluteness Arthur Apter buttons+switches CH chess computability continuum hypothesis countable models definability determinacy elementary embeddings forcing forcing axioms games GBC generic multiverse geology ground axiom HOD hypnagogic digraph indestructibility infinitary computability infinite chess infinite games ITTMs kids KM large cardinals maximality principle modal logic models of PA multiverse open games ordinals Oxford philosophy of mathematics pointwise definable potentialism PSC-CUNY supercompact truth universal algorithm universal definition universal program Victoria Gitman
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