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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: definability

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Pointwise definable models of set theory, extended abstract, Oberwolfach 2011

Posted on January 9, 2011 by Joel David Hamkins
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[bibtex key=Hamkins2011:PointwiseDefinableModelsOfSetTheoryExtendedAbstract]

This is an extended abstract for the talk I gave at the Mathematisches Forschungsinstitut Oberwolfach, January 9-15, 2011.

Slides | Main Article

 

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Posted in Publications, Talks | Tagged definability, forcing, HOD | Leave a reply

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Infinitely More

How we might have viewed the continuum hypothesis as a fundamental axiom necessary for mathematics

By mounting a philosophical historical thought experiment, I argue that our attitude toward the continuum hypothesis could easily have been very different than it is.

Joel David Hamkins
3 HR AGO
3
2
Take my Philosophy and Logic of Games final exam!

Can you pass the exam for my games course?

Joel David Hamkins
May 14
8
10
Pushpast

Can the triangles push past the circles?

Joel David Hamkins
May 7
1
Proof and the Art of Mathematics, MIT Press, 2020

Recent Comments

  • David Roberts on Skolem’s paradox and the countable transitive submodel theorem, Leeds Set Theory Seminar, May 2025
  • David Roberts on A potentialist conception of ultrafinitism, Columbia University, April 2025
  • Joel David Hamkins on A potentialist conception of ultrafinitism, Columbia University, April 2025
  • David Roberts on A potentialist conception of ultrafinitism, Columbia University, April 2025
  • David Roberts on A potentialist conception of ultrafinitism, Columbia University, April 2025

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  • Comment by Joel David Hamkins on The club filter in definable preorders
    Well, that isn't really correct, since ω1+1 embeds into those ωn. But there is a version of the question, I suppose, that drops that hypothesis, and this is still interesting. Under AD, Jackson has investigated which cardinals are measurable, but I am unsure whether we know the cofinalities of the cardinals. An affirmative answer to […]
  • Comment by Joel David Hamkins on The club filter in definable preorders
    Under AD, we know ω1 and ω2 are measurable, via the club filter, and ωn is not measurable $3\leq n
  • Comment by Joel David Hamkins on Is there a general way to translate well-founded and non-well founded models of theories?
    It is the same with Boffa, which has many automorphisms, although one needs parameters to define the automorphisms.
  • Comment by Joel David Hamkins on Is there a general way to translate well-founded and non-well founded models of theories?
    The whole universe. For example, if we add the axiom that there are exactly two Quine atoms, and everything else is generated from them in a well-founded hierarchy, then swapping them is a definable automorphism of the universe.
  • Comment by Joel David Hamkins on Is there a general way to translate well-founded and non-well founded models of theories?
    That won't be true, since ZFC-Reg+exists ill-fdd has extensions with definable automorphisms, and these can never be bi-interpretable with ZFC, which is definably rigid.
  • Comment by Joel David Hamkins on Is there a general way to translate well-founded and non-well founded models of theories?
    Yes, that is what I meant. For example, Aczel's anti-foundational theory (with choice) is bi-interpretable with ZFC.
  • Comment by Joel David Hamkins on Is there a general way to translate well-founded and non-well founded models of theories?
    Another way to answer your question: ZFC-Reg has extensions that are bi-interpretable with ZFC, and if we are working in such an extension, then every model of any fragment of ZFC can be seen from the interpreted ZFC point of view to be such a model. This provides a way of translating back and forth […]
  • Comment by Joel David Hamkins on Is there a general way to translate well-founded and non-well founded models of theories?
    Of course that is not provable in ZFC-Reg, since by the incompleteness theorem it is consistent with ZFC-Reg that there are no models of ZFC-Reg at all, let alone a well-founded model.

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