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1d
comment Why should “small” P be preferred?
I almost agree with @PerAlexandersson; I'd just interchange two words in his comment: I am entirely not clear what the question is.
2d
comment Two rational and one irrational root of a cubic?
Or look at the quadratic term.
2d
comment Making idempotent element by a relation
@YemonChoi Thanks, but I think I fixed it now. When I went to edit the answer, I found that all of the TeX formatting that I had originally typed had disappeared --- not only the dollar signs but also macros like mathbb. I have no idea why, but I think I've now put it all back the way it was, and I hope it stays that way.
2d
revised Making idempotent element by a relation
added 70 characters in body
2d
comment Making idempotent element by a relation
Well, the TeX became unreadable in the answer also. The symbols immediately after b and c are subscripts; what follows x and y are exponents; in particular, the last x in the answer is supposed to have 2m as an exponent.
2d
answered Making idempotent element by a relation
2d
comment “Graph Individualization”[ reference request]
It seems that the result of your individualization process depends very sensitively on the order in which vertices are listed, e.g., which vertex serves as $v_n$ to begin the process. Under these circumstances, I don't see the relevance of this to the graph isomorphism problem, and therefore I doubt that the process has been studied, unless it was for an entirely different purpose.
Aug
29
answered Notation: $Sigma$ and $Pi$ of intersections
Aug
28
comment How do mathematicians find the underlying idea?
Simon Henry wrote "maybe draw a picture"; I'll omit the "maybe". The way to understand this proof is to visualize what the inequalities say about distances.
Aug
28
comment How big is the lattice of all functions?
I'm afraid I can't verify the "not hard to verify" claim in the question. If I delete one element, say $q$, from $A$, then the value of $f_A(n)$ decreases by $1$ for all $n\geq q$. So the new and old $A$'s represent the same element of $\mathcal P(\omega)/$fin, but the new and old $f_A$'s represent different elements of $\mathcal L$.
Aug
27
comment Why should we care about “higher infinities” outside of set theory?
I don't understand the statement "... suppose we did not have the PowerSet Axiom, One could still prove that the reals had a higher cardinality than the naturals ...." Without the power set axiom, you don't know that there is a set of all reals. Are you working in a theory with proper classes so that you can still talk about the cardinality of the class of reals?
Aug
27
comment Does order-preserving equal continuous?
In view of the comments at the linked question, this interval topology agrees with the usual topology in the case $P=Q=\mathbb R$, and here there are trivial counterexamples for both of the proposed implications.
Aug
23
comment What are key $\Sigma^0_2$ or $\Pi^0_3$ theorems?
Although I'm not a number theorist and therefore can't immediately give you details, I believe that there are some important results saying that certain Diophantine equations have only finitely many solutions, without giving an explicit bound for those solutions. The natural formalization of such a statement would be a $\Sigma^0_2$ sentence.
Aug
22
answered Differential geometry without the Hausdorff condition or the second axiom of countability
Aug
20
comment How many rearrangements must fail to alter the value of a sum before you conclude that none do?
Maybe I'm just being dense (I'm still a little jet-lagged) but at the end of Part I of the proof, I don't see why $a_{n+1}>f_b(a_n)$ implies $b(a_{n+1})>b(a_n)$. I see that it implies that $b$ can't map $a_n$ up to $a_{n+1}$ and can't map $a_{n+1}$ down to $a_n$, but why can't it map $a_n$ up and $a_{n+1}$ down, to some in-between location where they're out of order? (If I'm not just being dense and this is really a problem, it can clearly be solved by modifying the definition of $f_b$, so the theorem survives.)
Aug
20
comment Approximation of Borel sets by a countable collection of majorants
Your argument shows a little more than you said, namely that you can't get every Borel $E$ of positive measure to contain some $E_n$ modulo measure zero sets. This too can be done directly. Given any proposed $E_n$'s of positive measure, pick subsets $G_n$ of positive but sufficiently small measure so that $E=I-\bigcup_nG_n$ has positive measure.
Aug
20
answered Approximation of Borel sets by a countable collection of majorants
Aug
16
comment What are some reasonable-sounding statements that are independent of ZFC?
@ToddTrimble You've been listening to set theorists more than topologists. (Admittedly, the topologists who are interested in cellularity are often set theorists in disguise.)
Aug
16
answered Looking for the name of a mathematical symbol that looks remotely like 1 (answer: indicator function)
Aug
15
comment addition on an affine scheme
The addition map is induced by the ring homomorphism from $K[x_1,\dots,x_n]$ to $K[x_1,\dots,x_n]\otimes K[x_1,\dots,x_n]\cong K[x'_1,\dots,x'_n,x''_1,\dots,x''_n]$ that sends each $x_i$ to $x'_i+x''_i$.