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Tag Archives: Sean Carroll

Puzzles of reality and infinity, Mindscape Podcast

Posted on July 15, 2024 by Joel David Hamkins

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I was interviewed by Sean Carroll for his Mindscape Podcast, broadcast 15 July 2024.

282 | Joel David Hamkins on Puzzles of Reality and Infinity

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Posted in Talks, Videos | Tagged Gödel, incompleteness, Mindscape, multiverse, philosophy of mathematics, Sean Carroll | Leave a reply

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Mathoverflow activity

Comment by Joel David Hamkins on Ultralimit versus partial limit
In my view it is inadvisable to use the symbol $\omega$ in a set-theory context to mean something other than the first infinite ordinal. This invites confusion, and it will cause problems as soon as one wants to refer to the ordinal $\omega$. It's like using $e$ to denote some other real number in a […]
Comment by Joel David Hamkins on Infinite hash function $h:\{0,1\}^\omega\to \{0,1\}^*$
The constant all-zero function seems to fulfill your desired property. Have you asked what you intended?
Comment by Joel David Hamkins on Examples of ubiquitous objects that are hard to find?
@none Yes, that seems to be exactly the same idea.
Comment by Joel David Hamkins on Is every nowhere dense closed set contained in the boundary of a regular open set?
@ChayimLowen I had been worried about both objections, but I see now how the other objection is resolved.
Comment by Joel David Hamkins on Is every nowhere dense closed set contained in the boundary of a regular open set?
I don't see that the proof of the claim works, even for the suggested modification. Suppose for example $X$ consists of a rapidly converging sequence, with its limit. The $1/i$ requirement is too lax to obtain the "exactly $X$" claim, since I could choose $c_i$ quite freely in this event.
Answer by Joel David Hamkins for Examples of ubiquitous objects that are hard to find?
For each natural number $k$, almost all finite strings have a Kolmogorov complexity at least $k$, that is, they are not the output result of a program of size less than $k$. But in light of the Chaitin incompleteness theorem, for all sufficiently large values of $k$, and indeed beginning with comparatively small values of […]
Comment by Joel David Hamkins on Examples of ubiquitous objects that are hard to find?
Sure. Or "arithmetic", "projective", "Borel". There are many degrees of complexity and definability.
Comment by Joel David Hamkins on How many subsets of $\omega^\omega$ are isomorphic to $\omega^\omega$?
@bof The notation $\omega^\omega$ is also commonly used the way Dominic is using it. This is a standard notation for Baire space.

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