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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: The Church of Logic

The Church of Logic podcast, April 2025

Posted on April 21, 2025 by Joel David Hamkins
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I was interviewed by Cody Roux for The Church of Logic podcast—a fascinating sweeping conversation on issues in the philosophy of mathematics and set theory, including what I described as a fundamental dichotomy between two perspectives on the nature of mathematics and what it is all about. Cody and I have affinities with opposite sides of this dichotomy, which made for a fruitful exchange.

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Posted in Talks, Videos | Tagged Cody Roux, philosophy of logic, philosophy of mathematics, philosophy of set theory, podcast, The Church of Logic | Leave a reply

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
May 22
5
12
Take my Philosophy and Logic of Games final exam!

Can you pass the exam for my games course?

Joel David Hamkins
May 14
10
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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  • Answer by Joel David Hamkins for How might fundamental mathematics differ for entities with intuitive comprehension of the continuum?
    The answer to your question is the subject of descriptive set theory, which is all about trying to understand the hierarchy of logical complexity that arises in a context where the real numbers are given as basic objects. This is far beyond the arithmetic hierarchy, studying the projective hierarchy, and we analyze the complexity of […]
  • Comment by Joel David Hamkins on How does a global well order provide a selector?
    It is definable from
  • Comment by Joel David Hamkins on A form of reverse mathematics that works with hereditarily finite sets instead of numbers
    Ah, sorry, I don't have a reference. But I think of this as a trivial step of applying the bi-interpretation.
  • Comment by Joel David Hamkins on How might fundamental mathematics differ for entities with intuitive comprehension of the continuum?
    I was writing an answer to this question, all about descriptive set theory, but the question was closed.
  • Answer by Joel David Hamkins for How does a global well order provide a selector?
    I have a blog post about exactly this kind of issue. See The global choice principle.
  • Answer by Joel David Hamkins for A form of reverse mathematics that works with hereditarily finite sets instead of numbers
    I don't have specific references for you, but it is basically immediate to translate between these two perspectives, which are bi-interpretable via the Ackermann encoding. Given numbers and arithmetic, we can define nEm if and only if the nth binary bit of m is 1, and this makes ⟨N,E⟩ isomorphic to ⟨HF,∈⟩. Conversely, […]
  • Answer by Joel David Hamkins for Universal sequence u:N→N with respect to subsequences
    Yes, just let u mention every number infinitely many times. u=⟨0,0,1,0,1,2,0,1,2,3,…⟩ Given any sequence f, you can define s(n) to pick out the index of the next occurence of f(n) inside u, and so f=u∘s as desired.
  • Comment by Joel David Hamkins on Is existence of one step downshifting embeddings consistent with Stratified ZF?
    But also, this notion of embedding is very weak. Indeed, ZFC proves that there are embeddings j:V→V that are definable and not the identity. Being definable, they can appear in the replacement and separation axioms. I prove this in my paper: worldscientific.com/doi/abs/10.1142/S0219061313500062. But those j will not have j(α)+1=α, and indeed, they don't even […]

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