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Joel David Hamkins

mathematics and philosophy of the infinite

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Recent Comments

Ali Enayat on Gödel incompleteness, graduate course, Notre Dame, Fall 2024
zencv on Gödel incompleteness, graduate course, Notre Dame, Fall 2024
Joel David Hamkins on Gödel incompleteness, graduate course, Notre Dame, Fall 2024
Ali Enayat on Gödel incompleteness, graduate course, Notre Dame, Fall 2024
Joel David Hamkins on Gödel incompleteness, graduate course, Notre Dame, Fall 2024

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Mathoverflow activity

Comment by Joel David Hamkins on What is the extent of a $\Sigma$-product of a (uncountable) power of a (countable) discrete space?
@StevenClontz Thanks for that. But in truth, I find that usage very odd.
Comment by Joel David Hamkins on What is the extent of a $\Sigma$-product of a (uncountable) power of a (countable) discrete space?
This looks helpful: matstud.org.ua/ojs/index.php/matstud/article/view/274
Comment by Joel David Hamkins on What is the extent of a $\Sigma$-product of a (uncountable) power of a (countable) discrete space?
Should it be $\sigma$-product rather than $\Sigma$-product, since it means the countable-support product? This would align with $\sigma$-algebra, $\sigma$-additivity, and so forth.
Comment by Joel David Hamkins on Examples of concrete games to apply Borel determinacy to
I find it natural also to consider the game where the players determine which (finite) part of the atomic diagram that are revealing at each turn, rather than having this determined by parity as in your description. This makes it something like the EF games, and also similar to the truth-telling game, where the interrogator […]
Comment by Joel David Hamkins on Examples of concrete games to apply Borel determinacy to
Your first example isn't necessarily just about graphs, but really the truth of any $\mathcal{L}_{\omega_1,\omega}$ sentence in a countable model of any signature type. I imagine the players collectively defining the atomic structure of the model, with Alice trying to make the sentence true and Bob trying to make it false. This way of thinking […]
Comment by Joel David Hamkins on Examples of concrete games to apply Borel determinacy to
Yes, it was an open question whether it is Borel, and the answer is nontrivial. And yet, the only reason we know in ZFC that infinite Hex is determined in the general case is because of Borel determinacy, so it seems relevant.
Answer by Joel David Hamkins for Examples of concrete games to apply Borel determinacy to
The game of infinite Hex, proceeding from an arbitrary position, is a good example with all the features you seek. The game was the subject of my Oxford student Davide Leonessi's masters MFoCS dissertation work, and we give an introduction in our article: Joel David Hamkins, Davide Leonessi, Infinite Hex is a draw, Integers 23, […]
Comment by Joel David Hamkins on why $L=\{\langle M\rangle\mid M \text{ is a TM that accepts all even number}\} \notin \text{RE}$
In particular, it is also incorrect to say that $L$ is co-RE, since that would make it complexity $\Pi^0_1$, but it is $\Pi^0_2$.

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