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bio website math.lsa.umich.edu/~ablass
location US
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visits member for 4 years, 8 months
seen 11 hours ago

Dec
22
comment Removing subtrees
In the $[0,1]$ model, condition 1 requires the set to be a closed set with empty interior, but it can still have positive Lebesgue measure.
Dec
20
comment Complete resolutions of GCH
In Solovay's theorem the function $F$ should be reasonably absolute. Otherwise, there is the silly function defined by setting $F(\kappa)=(2^\kappa)^+$ for which you certainly can't have $2^\kappa=F(\kappa)$. This silly $F$ is definable but it's not absolute and, when you try to match it with $2^\kappa$, it "runs away" so that you never succeed.
Dec
18
comment When does Skolemization require the axiom of choice?
By referring to the statement $(\forall x\exists y\,\phi(x,y))\iff(\exists f\forall x\,\phi(x,f(x))$ as an identity, the OP probably meant that this statement should be valid, i.e., true in all structures. If that was the intention, then, as Joel said, the full axiom of choice is required.
Dec
16
comment Producing finite objects by forcing!
Wouldn't your argument imply that every ill-founded recursive tree has a hyperarithmetical path? And doesn't that contradict a theorem of Kleene? I think the best you can expect in general is a path recursive in a complete $\Pi^1_1$ set. (In the case at hand, though, the Kleene-Post construction seems to produce $\Delta^0_2$ examples.)
Dec
15
comment Producing finite objects by forcing!
By "a more refined absoluteness argument", do you mean getting absoluteness from some forcing model down to a model in which all subsets of $\omega$ are c.e.? That would seem to be too optimistic since the small model wouldn't be closed under taking complements of subsets of $\omega$. Another way to say that is that it's hard to imagine a Cohen real as looking like a c.e. set in any useful way.
Dec
15
comment Producing finite objects by forcing!
On the one hand, I don't ordinarily say "priority" when all the requirements are compatible, so no prioritization is needed. On the other hand, I suppose the hierarchy "finite injury, infinite injury, monster, ..." could be extended backward to "0 injury", which would be Kleene-Post. As for making "genuine" priority arguments look like forcing, I think there have been many attempts over the years, but I haven't really learned about any of them. Some (surely not all) of the relevant names are Yates, Lachlan, and Lerman.
Dec
15
comment Producing finite objects by forcing!
Although you mentioned a priority construction, I think your last paragraph implies that priority isn't really involved. Won't "the construction of sufficiently generic degrees" amount to a Kleene-Post argument? (Priority would presumably become relevant if you wanted to do this with c.e. degrees.)
Dec
13
awarded  Nice Answer
Dec
13
awarded  co.combinatorics
Dec
12
answered Lots of combinatorial interpretations of Catalan numbers
Dec
11
comment Around Continuum Hypothesis
@JesseElliott Easton's results let you make $2^{\aleph_\alpha}$ the $\alpha$-th weakly inaccessible for all $\alpha$ such that $\aleph_\alpha$ is regular, but not for singular $\aleph_\alpha$. Getting anything like this for singular cardinals will require considerable large-cardinal strength, if it isn't outright inconsistent.
Dec
11
comment Around Continuum Hypothesis
@JesseElliott After the forcing, $\kappa$ is no longer strongly inaccessible, but it remains weakly inaccessible, because weak inaccessibility is defined using only the notions of cardinal and cofinality, neither of which changes when the forcing notion satisfies the countable chain condition.
Dec
11
comment Interweaving two indexed families of filters
I suppose there are two tools that I use. One is that I've been thinking about filters since approximately 1968, so I've developed some intuition about what is or isn't likely to work. The other, which I used in this case, is to try to carefully prove your conjecture, and see where the proof runs into difficulty. That provided a hint of how to build a counterexample, taking advantage of the difficulty.
Dec
10
answered Interweaving two indexed families of filters
Dec
8
answered First-order definable bijection between $P(On)$ (or $No$) and $V$? (Is this equivalent to $V = HOD$?)
Dec
7
answered CCC Forcing and $\omega_1$ conditions
Dec
5
comment Link between abelian groups and endomorphisms
I'm not going to fix the typo "principal idea domain" because I sort of like it as it is.
Nov
30
comment How can we join two points with a small ruler?
@PeteL.Clark I think the usual notion of Euclidean construction allows a certain flexibility in doing "arbitrary" things. For example, to bisect an angle, I'd start by drawing a circular arc, centered at the angle's vertex, and meeting both of the angle's sides. The radius of this arc is arbitrary, and there seems to be no way to make it specific unless we're given somewhat more than just the angle to be bisected (meaning its vertex and two rays).
Nov
24
awarded  Quorum
Nov
21
comment how to reduce 3-colorable graph to this?
I don't think it's a good idea to use the set of vertices of the graph as the X in your problem. The input to 3-coloring (namely the graph) is only polynomially larger than the set of vertices, but the input to your question, including the family S, could be exponentially larger than X. So "PTime" for your problem could be exponential time relative to the size of X.