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Visiting assistant professor at Miami University of Ohio.


Jul
19
accepted Partitions of the Cantor space into parities
Jul
19
awarded  Nice Question
Jul
17
comment Partitions of the Cantor space into parities
@Asaf, a principal ultrafilter won't produce a parity partition in the way illustrated above, since if $\{n\}\in U$ then flipping any bit past the $n$th one won't swap the elements of $A$ and $B$.
Jul
17
answered Partitions of the Cantor space into parities
Jul
15
comment Partitions of the Cantor space into parities
Yes, or you could even show that for all $x,y\in A$ there is $z\in A$ such that $E(z) \subseteq E(x)\cap E(y)$.
Jul
15
revised Partitions of the Cantor space into parities
Added 4th remark, fixed types
Jul
15
comment Partitions of the Cantor space into parities
Good question: I'll add another remark explaining what's clear and what's not clear in that strategy.
Jul
15
comment Partitions of the Cantor space into parities
@AsafKaragila, that's a good suggestion, thanks!
Jul
15
revised Partitions of the Cantor space into parities
improved formatting
Jul
14
asked Partitions of the Cantor space into parities
Jul
11
comment cov(meager) strictly between $\aleph_1$ and $2^{\aleph_0}$
I think the ideas from Bartoszynski and Judah should lead to a proof. If you want another reference to look at, Blass's chapter of the Handbook of Set Theory should tell you all you need to know. (And I think it's freely available on his webpage.)
Jul
10
comment Construction of Ultrafilters
@Norbert: Apparently I am also guilty of perpetuating false information; the existence of a nonprincipal ultrafilter is weaker than the Boolean Prime Ideal Theorem, which implies that every filter can be extended to an ultrafilter. BPIT is still weaker than full AC, though.
Jul
10
comment Construction of Ultrafilters
I also feel obligated to point out that a model of "all sets of reals are Lebesgue measurable" requires the consistency of an inaccessible cardinal, whereas "all sets of reals have the BP" doesn't have any large cardinal strength.
Jul
10
comment Construction of Ultrafilters
The last bit, namely that an ultrafilter can't be measurable or have the Baire property, doesn't take too much work; the construction of the Solovay model, on the other hand, requires a decent background in forcing, as far as I know.
Jul
10
comment Construction of Ultrafilters
@KyleGannon: In answer to the new question, you could look at the Solovay model, which is a well-known model of ZF + DC + "all sets of reals are Lebesgue measurable and have the Baire property". A nonprincipal ultrafilter cannot be measurable or have the Baire property.
Jul
10
comment Construction of Ultrafilters
@Norbert: it's equivalent to a weak form of AC, namely the Boolean Prime Ideal Theorem.
Jul
3
comment Is the class of countable posets well-quasi-ordered by embeddability?
Great answer, thanks!
Jul
3
accepted Is the class of countable posets well-quasi-ordered by embeddability?
Jul
2
awarded  Curious
Apr
3
answered Different definitions of P-Points (ultrafilters)