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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: NWO

Modal logics in set theory, NWO grants, 2006 – 2008

Posted on April 30, 2006 by Joel David Hamkins
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Modal logics in set theory, (with Benedikt Löwe), Nederlandse Organisatie voor Wetenschappelijk (B 62-619), 2006-2008.

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Posted in Grants and Awards | Tagged Amsterdam, NWO | Leave a reply

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Proof and the Art of Mathematics, MIT Press, 2020

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Recent Comments

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  • Comment by Joel David Hamkins on Computably constructing a set that is not in a free ultrafilter
    I have in mind various priority argument constructions, but none yet seem to work.
  • Comment by Joel David Hamkins on Computably constructing a set that is not in a free ultrafilter
    In light of @AlexKruckman's comment, we needn't involve ultrafilters at all. The question is really asking: is there a uniform computable process S:f↦S(f) obeying the OP's cardinality constraint on total computable increasing f, such that the family of these S(f)s has the FIP? This is precisely what it takes to have a counterexample.
  • Answer by Joel David Hamkins for Formal theory of structure
    To my way of thinking, all of mathematics is concerned at bottom with structure—to undertake mathematics is to introduce a study a particular kind of structure, whether it is graphs in graph theory, groups in group theory, topologies in topology, and so on. In this light, every foundational theory of mathematics—set theory, type theory, category […]
  • Comment by Joel David Hamkins on About definition of proper forcing
    Yes, I agree. We just quantify over countable M. Of course, in ZFC when we say that something is true, what we mean is that it is true in the ambient set theoretic realm that the theory is talking about. And that realm can be called V, referring to the current universe under consideration.
  • Answer by Joel David Hamkins for About definition of proper forcing
    Yes, countable in V. Properness of the forcing notion P is something that is verified in the ground model V over which you intend to force.
  • Comment by Joel David Hamkins on References for incompleteness proofs using infinite trees or König's lemma
    Yes, there are diagonal arguments almost everywhere in logic.
  • Comment by Joel David Hamkins on References for incompleteness proofs using infinite trees or König's lemma
    One could object that in order to provide the tree, one must essentially have already proved the undecidability of the halting problem, and then one could prove the incompleteness theorem directly with that, using proofs to decide the halting problem, rather than doing the further work to create the computable tree with no computable branch. […]
  • Answer by Joel David Hamkins for References for incompleteness proofs using infinite trees or König's lemma
    I'm not sure about the particular proof your professor has in mind, but here is a proof using trees and paths-through-trees. First, we prove the classic result in computability theory that there is a computable infinite tree $T\subset 2^{

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