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Joel David Hamkins

mathematics and philosophy of the infinite

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Infinitely More

The tactical variation of the fundamental theorem

We prove the tactical variation of the fundamental theorem of finite games—for finite games with sufficiently rich board positions, one of the players has a winning tactic or both have drawing tactics

Joel David Hamkins

Tactics versus strategies in the theory of games

How do tactics differ from strategies? Does the fundamental theorem of finite games hold for tactics? Must every finite game have a winning tactic for one player or drawing tactics for both?

Joel David Hamkins

The curvature of space

An excerpt from Lectures on the Philosophy of Mathematics

Joel David Hamkins

Buy Me a Coffee

Recent Comments

Lecture series on the philosophy of mathematics | Joel David Hamkins on Lectures on the Philosophy of Mathematics
How the continuum hypothesis might have been a fundamental axiom, Lanzhou China, July 2025 | Joel David Hamkins on How the continuum hypothesis could have been a fundamental axiom
Joel David Hamkins on Lectures on Set Theory, Beijing, June 2025
Joel David Hamkins on Lectures on Set Theory, Beijing, June 2025
Mohammad Golshani on Lectures on Set Theory, Beijing, June 2025

JDH on Twitter

Mathoverflow activity

Comment by Joel David Hamkins on Algorithms to count restricted injections
Have you mixed up $𝑛$ and $𝑚$ in the beginning? You have $𝑓 (𝑎)$ , where $𝑎$ is from ${1, \dots, 𝑚}$ , but the domain of $𝑓$ is said to be ${1, \dots, 𝑛}$ . If $n
Comment by Joel David Hamkins on Terminology: commonly used name for an $𝜔$ machine?
Of course ultimately the computational power has nothing to do with fitting the computation into finite time, but rather just the idea of making sense of a computation with infinitely many steps. So you may be interested in the Büchi automata (en.wikipedia.org/wiki/B%C3%BCchi_automaton), and beyond this, the infinite time Turing machines (jstor.org/stable/2586556), which extend the operation […]
Comment by Joel David Hamkins on Is this theory of bottomless hierarchy, consistent?
Ah, sorry, you had the comma before $\to$ not after, namely, $\exists 𝑦 \in 𝐴 : 𝜙, \to$ .
Comment by Joel David Hamkins on Is this theory of bottomless hierarchy, consistent?
I confess that I have long been a little confused by your manner of using colons and commas in formal expressions, since it is different from what I am used to in first-order logic or in type theory. For example, how am I to read the meaning of " $\exists 𝑦 \in 𝐴 : 𝜙 \to,$ "? And how are we […]
Comment by Joel David Hamkins on Is every external downshifting elementary embedding $𝑗$ with $𝑗 (𝑥) = 𝑗 [𝑥]$ , an automorphism?
Ah, yes, of course. Thanks!
Answer by Joel David Hamkins for About forcing method
Yes, part of your perspective is correct—we can make sense of forcing over any model of set theory. We can in effect internalize the concepts of forcing and express everything we need inside ZFC rather than in the metatheory. The assertion of " $𝜑$ is forceable", meaning that it is true in some forcing extension, is […]
Comment by Joel David Hamkins on Is it possible to transform a statement of unsolvabilty to an equivalent one by using a bounded universal quantifier
OK, I have posted the argument I had in mind for the multi-variable case.
Answer by Joel David Hamkins for Is it possible to transform a statement of unsolvabilty to an equivalent one by using a bounded universal quantifier
Let me answer negatively for the case where the polynomial $𝑝$ is a polynomial in several variables $𝑝 (𝑥 1, \dots, 𝑥 𝑛)$ over the integers. To begin, for any given program $𝑞$ , consider the c.e. set $𝐸 𝑞$ that undertakes the algorithm of checking whether $𝑞 (0)$ halts, then whether $𝑞 (1)$ halts, then whether $𝑞 (2)$ halts, and so forth, and each […]

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