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3h
comment Projective family of probability spaces
Liviu is right. In a category of probability spaces, one wants morphisms that respect the whole probability space structure, not just the $\sigma$-algebra but also the measure.
9h
comment Antichain on $\mathcal{P}(\omega)/fin$ of cardinality $2^{\aleph_0}$?
It might be worth pointing out that this example works whether you use the set-theorist's definition of antichain (pairwise incompatible) or everybody else's definition (pairwise incomparable). If you wanted only the latter, weaker sort of antichain, you could take the sets $\{k:r<q_k<r+1\}$ for all reals $r$.
9h
comment Publication in proceedings
@SébastienPalcoux The word "Proceedings" in the title of a journal tells me nothing about the quality of the Journal. Proc. Amer. Math. Soc. is a high quality journal. (Full disclosure: Some years ago, I was on its editorial board.) Proc. Indian ... might also be a good journal, but I'd have to check; I never heard of it until just now, and that makes me a little suspicious.
2d
awarded  Nice Answer
Mar
27
comment In set theory, is there a name for a function which maps the empty set to zero and all the others to one?
It's not clear what you want, as you've already defined $f$. Do you want a definition that "looks more set-theoretic"? Do you want one that doesn't use a case distinction? Or what? One possible description of your $f$ is that $f(S)=$ the number of functions to the empty set from the set of functions to the empty set from $S$, i.e., $0^{0^{|S|}}$. Is this the sort of thing you were looking for?
Mar
27
comment A property of the Frechet filter and every ultrafilter
You probably already have examples of filters lacking your property, but, just in case, here's one: For $i=0,1,2$, let $A_i=\{n\in\omega:n\equiv i\pmod3\}$. Let $\mathcal U$ and $\mathcal V$ be non-principal ultrafilters containing $A_0$ and $A_2$, respectively, and let $\mathcal F=\mathcal U\cap\mathcal V$, so $\mathcal F^+=\mathcal U\cup\mathcal V$. I claim this $\mathcal F$ does not have the property you described. The counterexample is $A=A_0\cup A_1$ and $B=A_0\cup A_2$. Then $A\cap B=A_0$ and $\bigcup\{[n,\text{next}(A,n)]:n\in A\cap B\}=A_0\cup A_1\notin\mathcal F$.
Mar
26
comment A property of the Frechet filter and every ultrafilter
Does "not the above under a finite-to-one map" mean that the filter you seek should not have the Frechet filter or an ultrafilter as a finite-to-one image? So, in particular, you would require Filter Dichotomy to fail?
Mar
24
comment Formal definition of arithmetic transfinite recursion
Emil is right about Simpson's book. In more detail: Stephen G. Simpson, "Systems of Second-Order Arithmetic"
Mar
21
comment What algebraic identities does the iteration of forcing operation satisfy?
The definition of $G$ should probably require that ZFC proves $\exists x\,\phi(x)$, because otherwise your $\odot$ operation won't be single-valued.
Mar
15
comment Independence of the countable axiom of choice
In the present situation, you can get from an urelement proof (the basic Fraenkel model) to a pure-set proof more easily than by the Jech-Sochor construction. Adjoin to the basic Fraenkel model an $A$-indexed family of Cohen reals (force with Fin($A\times\omega$,2)), and then take the pure part of the resulting model. You get the basic Cohen model, which, of course, violates countable choice.
Mar
14
revised What is the maximal number of sub spaces of a fixed dimension such that there is another sub space which intersects them are all null
pointed out that the question changed after this answer
Mar
14
comment $\mathfrak{p}=\mathfrak{b}=\mathfrak{a}=\aleph_1$ and $\mathfrak{d}=\mathfrak{c}=\kappa$
@Angel All these facts are in Section 11.3 of the handbook chapter cited in my answer. Unfortunately, you may need to look at some of the earlier sections of the chapter for some of the terminology used there. Although I don't have my copy of the Bartoszynski-Judah book "Set theory: On the structure of the real line" handy, I'm pretty sure all this information is there also.
Mar
13
answered $\mathfrak{p}=\mathfrak{b}=\mathfrak{a}=\aleph_1$ and $\mathfrak{d}=\mathfrak{c}=\kappa$
Mar
13
comment What is the maximal number of sub spaces of a fixed dimension such that there is another sub space which intersects them are all null
@var That seems to be a rather different question.
Mar
13
comment What is the maximal number of sub spaces of a fixed dimension such that there is another sub space which intersects them are all null
@EmilJeřábek You're right. Thanks. Now that you've pointed it out, there's an easier proof. Regard $V$ as $U\oplus Z$ from some $r$-dimensional $Z$, and note that the $W$'s that we need to count are just the graphs of linear maps from $Z$ to $U$. Such a map is given by an $(n-r)$ by $r$ matrix of elements of $\mathbb F_q$.
Mar
13
comment What is the maximal number of sub spaces of a fixed dimension such that there is another sub space which intersects them are all null
@var Well, if you have, as in the problem, $k$ "very general" subspaces $W_i$ and there exists a $U$ meeting them all at only $\{0\}$, then, for that $U$, your $k$ $W_i$'s would be among the spaces I counted, so $k$ would be at most the number, call it $A$, that I gave in my answer. Conversely, there are $A$ subspaces that satisfy your requirements, namely the $A$ spaces that I counted, because there's a $U$ meeting them all only at $\{0\}$. So, for a collection of $W$'s as in the question, the largest possible $k$ is $A$.
Mar
13
answered What is the maximal number of sub spaces of a fixed dimension such that there is another sub space which intersects them are all null
Mar
13
comment Brandt's definition of groupoids (1926)
I recall seeing groupoids called "Brandt groupoids" is some paper of Bill Lawvere, quite some time ago, but I can't find the reference now. So Bill was apparently aware of Brandt's contribution, even if the rest of us weren't.
Mar
11
comment Continuous functions and 2-bushy trees
$[T]$ is the set of infinite paths through the tree $T$, and the OP wants $f$ to be one-to-one or constant on such a set $[T]$.
Mar
11
comment Continuous functions and 2-bushy trees
I don't see why "restricted to any tree that branches on an even row, $f$ cannot be one-one." The reason is that, if $p$ and $q$ are the two immediate successors of a branch point $b$, the part of the subtree beyond $p$ may be entirely different from the part beyond $q$.