2d

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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 nonmeasurable cardinality is realcompact. 
2d

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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

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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

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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 
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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 
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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 
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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 $\lambda$?
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Nov
7 
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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 
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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 $\lambda$?
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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 
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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 rankintorank embeddings infinitely many times?
added 97 characters in body 
Oct
24 
asked  Can one take roots of rankintorank embeddings infinitely many times? 