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2d
comment Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?
There's probably a question about this on this site as well.
2d
comment Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?
Also math.stackexchange.com/q/393196/622 and other questions on the "Linked" menu to the right in that link.
2d
comment Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?
That is correct.
Feb
2
awarded  Necromancer
Feb
2
comment Axiom of countable choice need for the cantor-bernstein theorem
@Noah: It might be acceptable to ask this question on this site, I can certainly imagine one of the professors in my department running into me (or any other set theorist) in the hallway and asking this. Not like this, though. And some effort, like typing into the search box "Cantor Bernstein choice" and reading a bunch of the related results is more or less a prerequisite of asking anyway.
Jan
29
revised weak-* versus entropy growth
edited title
Jan
29
comment Proposals for polymath projects
Since I'm entirely unfamiliar with polymath projects, I was just idly wondering if progress which is non-effective is considered progress. Namely proving existence without giving any bounds or means for finding examples... (Particularly for the number theoretic or combinatorial problems)
Jan
28
comment Taller models of ZFC
@Carl: Yes, I'm just saying that the axiom "there is a proper class of $V_\alpha$'s which are models of ZFC" is weaker than the existence of an inaccessible. Of course there can be just one single worldly cardinal (cut the universe at the second worldly cardinal!). So all I'm saying is that you can get it from even milder assumptions. :-)
Jan
28
comment Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
@Salvo: I found a counterexample to the idea of using Gitik's model. I've added it to my answer, and at least for a while I will leave the answer here.
Jan
28
revised Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
added 2338 characters in body
Jan
28
comment Taller models of ZFC
@Carl: Just one inaccessible cardinal is enough. Since an inaccessible cardinal is the limit of worldly cardinals (those that $V_\alpha$ is a model of ZFC), so if $\kappa$ is inaccessible $V_\kappa$ satisfies that "There is a proper class of $\alpha$'s such that $V_\alpha$ is a model of ZFC". You can probably even have something weaker like $\kappa$ is a worldly limit of worldly cardinals.
Jan
27
comment Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
@Salvo: Let me think about it until the morning, and if I can't figure it out, I'll edit/delete my answer.
Jan
27
comment Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
@Salvo: I'm starting to think that if there is an atom which is not a singleton, then the measure cannot be extended (or rather if you can't choose from each atom a unique point). Because if singletons get $0$, then every set must have measure $0$.
Jan
27
comment Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
@Salvo: You're right, the argument is a bit off. I'll fix it later today! Thanks!
Jan
27
comment Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
@Salvo: The idea is due to Specker, yes. Gitik "merely" proved that it's also consistent.
Jan
26
revised Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
added 38 characters in body
Jan
26
comment Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
@Joel: You're probably right. But it will have to wait for the morning. :-) in any case, this is true in Gitik's model, and Arnie Miller gives some survey of this axiom (due to Specker if my memory serves me right) in his paper also.
Jan
26
comment Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
@Joel: In the rank function I defined here? It's not obvious, and Gitik proves this in the ultimate or penultimate section of his paper.
Jan
26
comment Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
@Joel: Define a rank function, singletons have rank $0$, and now step up with countable unions of sets of lower rank. Then in Gitik's model you get the entire universe like that. The point being that generally, sigma algebras are already more or less power sets to begin with in this model.
Jan
25
comment Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?
Sure. It's pretty late, so I might have had some serious oversight. Let me know if you notice anything fishy.