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.

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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

  • The surprising strength of reflection in second-order set theory with abundant urelements, CUNY Set Theory seminar, April 2022 | Joel David Hamkins on Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal
  • Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal | Joel David Hamkins on The surprising strength of reflection in second-order set theory with abundant urelements, CUNY Set Theory seminar, April 2022
  • Bokai Yao on The surprising strength of reflection in second-order set theory with abundant urelements, CUNY Set Theory seminar, April 2022
  • Jason Chen on The surprising strength of reflection in second-order set theory with abundant urelements, CUNY Set Theory seminar, April 2022
  • Quanta Magazine on Win at Nim! The secret mathematical strategy for kids (with challange problems in transfinite Nim for the rest of us)

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  • Comment by Joel David Hamkins on Are there interesting examples of theorems proved using ‘height’ extensions?
    In our paper, we prove essentially that all of the most common urelement theories are bi-interpretable with pure set theories. For example, ZFC with ZFCU + Ord many urelements or ℝ many, KM with KMU+omega many atoms, etc. A many for any class A of pure sets. We take this to explain on structuralist grounds […]
  • Answer by Joel David Hamkins for Are there interesting examples of theorems proved using ‘height’ extensions?
    Here is another example. The maximality principle in forcing is the scheme asserting of every statement $\varphi$ in the language of set theory that if there is forcing extension $V[G]$ of the set-theoretic universe $V$ for which all further forcing extensions $V[G][H]$ satisfy $\varphi$, then $\varphi$ was already true in the original universe $V$. The […]
  • Answer by Joel David Hamkins for Are there interesting examples of theorems proved using ‘height’ extensions?
    Here is another instance, which appears in my recent paper with Bokai Yao on second-order reflection in the context of KMU with abundant urelements. Joel David Hamkins and Bokai Yao, Reflection in second-order set theory with abundant urelemets bi-interprets a supercompact cardinal, 2022, arXiv:2204.09766. The following theorem is an immediate consequence of the main theorem. […]
  • Comment by Joel David Hamkins on Ordering of large cardinals by cardinality
    To my knowledge all those instances are still open.
  • Comment by Joel David Hamkins on Church-Turing tests and quasi-computational models
    My view, in keeping with my general attitude about the relation between mathematics and philosophy, is that one should undertake the philosophical task (which is how I had understood your question) in light of the related mathematical analysis. To analyze philosophically whether a given notion should count as computational, one should first know how it […]
  • Comment by Joel David Hamkins on Church-Turing tests and quasi-computational models
    Indeed, these hierarchies are intensely studied, and one gains much more refined information from them than just a yes/no answer as to whether the concept should count as computational.
  • Comment by Joel David Hamkins on Church-Turing tests and quasi-computational models
    @SamHopkins Yes, my answer is that one gains such insight by fitting it into the hierarchies I mention. This is how these hierarchies are often used.
  • Answer by Joel David Hamkins for Church-Turing tests and quasi-computational models
    Computability theory overflows with hierarchies of computability notions, which allow us precisely to compare the computational strength of diverse notions. We have the hierarchy of Turing degrees, which measures the strength of relative computability by oracles. We have the hierarchy of complexity theory and the complexity zoo, which measures the strength of diverse resource-limited computation. […]

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absoluteness Arthur Apter buttons+switches CH chess computability countable models definability determinacy elementary embeddings equivalence relations forcing forcing axioms games GBC geology ground axiom HOD hypnagogic digraph indestructibility infinitary computability infinite chess infinite games ITTMs Jonas Reitz kids KM large cardinals maximality principle modal logic models of PA multiverse open games ordinals Oxford philosophy of mathematics potentialism PSC-CUNY supercompact truth universal definition universal program Victoria Gitman W. Hugh Woodin weakly compact
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