Joseph Van Name
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 2d comment Which Banach spaces are realcompact? Perhaps it is more illuminating to see the chain of implications from metrizability to realcompactness. It turns out that every metric space is paracompact. The paracompact spaces are precisely the spaces with a compatible supercomplete uniformity (supercomplete simply means that the hyperspace uniformity is complete). Every supercomplete uniform space is a complete uniform space. And below the first measurable cardinal, the realcompact spaces are precisely the topological spaces with a compatible complete uniformity. Therefore, every metric space of non-measurable cardinality is realcompact. 2d comment special extremally disconnected spaces with only finite isolated points Furthermore, the extremal disconnectedness of $[\kappa]^{<\omega}$ is equivalent to the Prikry lemma which states that for every statement $\sigma$ and every stem $s$ (the stems are the elements in the space $[\kappa]^{<\omega}$) there is some $A\in M$ where $(s,A)$ decides $\sigma$. I have not seen any papers or investigations where this relation between Prikry forcing and general topology has been seriously investigated before even though it seems very interesting. 2d comment special extremally disconnected spaces with only finite isolated points This is a really nice topological space which is very much related to Prikry forcing. Let's take a normal ultrafilter $M$ on a measurable cardinal $\kappa$ and define a topology on $[\kappa]^{<\omega}$ where a subset $U\subseteq[\kappa]^{<\omega}$ is open if and only if for each $(a_{1},...,a_{n})\in U$ there is some $A\in M$ where if $a\in A,a>a_{n}$, then $(a_{1},...,a_{n},a)\in U$ as well. Then $[\kappa]^{<\omega}$ is extremally disconnected, so the Boolean algebra of clopen sets of $[\kappa]^{<\omega}$ is a complete and it is the completion of the Prikry forcing. 2d comment Arguments against large cardinals Todd Trimble. Let me also mention that this obstruction to tameness at the first measurable cardinal also holds in topology for similar reasons. For example, every complete uniform space is realcompact if and only if there is no measurable cardinal. In particular, a discrete space is realcompact if and only if its cardinality is below the first measurable cardinal. Every extremally disconnected $P$-space is discrete if and only if there are no measurable cardinals. Also, every topological group is the fundamental group of a compact space if and only if there is no measurable cardinal. Nov 25 comment Cardinalities larger than the continuum in areas besides set theory As for my previous comment, a reference from the result about how $n$-hugeness suffices in the place of I3 is stated at least in Dougherty's paper "Critical points in an algebra of elementary embeddings." Nov 12 awarded Nice Question Nov 11 asked Do the Laver tables converge to the Sierpinski triangle with a line segment sticking out in the hyperspace topology? Nov 11 comment Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? Everett Piper. I was not familiar with argument yet. I will need to read up on the papers and notes that you have referenced. Nov 8 comment Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? Everett Piper. I am asking for an inaccessible cardinal $\lambda$ so that after forcing $\lambda$ becomes a singular strong limit cardinal of cofinality $\aleph_{0}$ and in the generic extension there is an elementary embedding from $V_{\lambda}$ to $V_{\lambda}$ which may be possible since $\lambda$ is after forcing a strong limit cardinal of countable cofinality. Are the questions that I asked clearly stated or should I reword them? Nov 7 revised Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? added 96 characters in body Nov 7 comment Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? Victoria. Yes. We force the cofinality $\lambda$ to be $\aleph_{0}$. Nov 7 comment Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? Asaf. I meant that in $V[G]$ there is an I3 elementary embedding from $V_{\lambda}^{V[G]}$ to $V_{\lambda}^{V[G]}$. Nov 7 comment Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? Victoria. I meant $V_{\lambda}^{V[G]}$ to make it look like an I3 and a I1 cardinal. Asaf. I simply put the quotes there to specify the statement that $V[G]$ models, but I now removed them so that they do not cause confusion. I edited the question accordingly. Nov 7 revised Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? added 28 characters in body Nov 7 asked Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? Oct 26 revised How much energy will be released in the explosion when one shoots a superelastic billiard ball into a collection of still superelastic billiard balls? Edited according to Ben Crowell's comments. Oct 25 comment How much energy will be released in the explosion when one shoots a superelastic billiard ball into a collection of still superelastic billiard balls? Ben Crowell. Yes. One should use the center of mass frame. After all, after each bounce, the amount of kinetic energy should increase so that we have an explosion. And yes, all the laws of Newtonian physics except having coefficient of restitution greater than $1$ should be satisfied. Oct 25 asked How much energy will be released in the explosion when one shoots a superelastic billiard ball into a collection of still superelastic billiard balls? Oct 24 revised Can one take roots of rank-into-rank embeddings infinitely many times? added 97 characters in body Oct 24 asked Can one take roots of rank-into-rank embeddings infinitely many times?