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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Francesco Cavina

Mathematics, Philosophy of Set Theory and Infinity, Back to the Stone Age interview, May 2024

Posted on June 7, 2024 by Joel David Hamkins
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I was interviewed by Francesco Cavina for the Back to the Stone Age series on May 17, 2024, with a sweeping discussion of the philosophy of set theory, infinity, the continuum hypothesis, beauty in mathematics, and much more.

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Posted in Talks, Videos | Tagged continuum hypothesis, Francesco Cavina, infinity, philosophy of set theory | Leave a reply

Infinitely More

On going first

Would you rather go first or second? In many games, there is a definite advantage one way or the other. How can we redress these imbalances, if we seek to make truly fair and balanced games?

Joel David Hamkins
May 30
4
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
Proof and the Art of Mathematics, MIT Press, 2020

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  • Comment by Joel David Hamkins on Is subset-well-foundedness equivalent to the Axiom of Choice?
    Another subtle issue is that your formulation is not quite asserting that there is no infinite descending chain of cardinals, since by insisting that Xk+1⊆Xk you have in effect chosen an injection. So from your condition, I wouldn't necessarily know how to prove that there is no infinite descending sequence of cardinals.
  • Comment by Joel David Hamkins on Is subset-well-foundedness equivalent to the Axiom of Choice?
    Also, in your title you refer to well-orderedness, but already the linearity of cardinals is equivalent to the axiom of choice. Your question is closer to referring to well-foundedness, not well-orderedness, but even so, not quite, since the descending chain condition is not necessarily equivalent to well-foundedness without DC.
  • Comment by Joel David Hamkins on Is subset-well-foundedness equivalent to the Axiom of Choice?
    Note that your formulation is saying something like: there is no descending sequence of cardinals, but without DC this could be weaker than saying: the cardinals are well-founded, which would be: every nonempty sets of sets has one of minimal cardinality. For example, if there is an infinite Dedekind finite set, we can violate well-foundedness, […]
  • Comment by Joel David Hamkins on Is V[r0,…,rk,…] a proper extension
    If someone wants to push the idea through, please feel free to post an answer.
  • Comment by Joel David Hamkins on Is V[r0,…,rk,…] a proper extension
    Yes, in fact I was thinking about exactly that case. One needs to verify that the composition of those forcing notion is still improper. Is this clear?
  • Comment by Joel David Hamkins on Is V[r0,…,rk,…] a proper extension
    This would answer your question negatively, if we had an improper extension that was stationary preserving and generated by a real (with a Cohen real present). Perhaps we can just do some improper forcing stat-preserving forcing, followed by almost disjoint coding to get the extension generated by a real?
  • Comment by Joel David Hamkins on Is V[r0,…,rk,…] a proper extension
    If you have any Cohen real at all and W is generated by some real r, not necessary Cohen, then you can achieve the situation of your third bullet point, by splitting the Cohen real into countably many, and changing one bit in each so as to code the given real r into the sequence.
  • 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 […]

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