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5h

comment 
If a result is apparently provable with AC, is actually independent of ZF?
To say that it was first obtained by Lindenbaum and Tarski in 1926 is to skip over the actual history of the situation. Certainly it was not published in 1926, and Tarski many years later said he couldn't remember how Lindenbaum's argument for a key lemma went, but found a proof for it himself. The (Conway)Doyle paper describes all this better than I can in a small comment box (they, or he [Doyle], surmise that they hit upon the argument Lindenbaum must have used, but I'd treat that as just a guess). 
7h

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Embedding finite lattices into the lattice of partitions of a finite set
I realize. I'll leave my answer in case it's still useful. 
7h

answered  Embedding finite lattices into the lattice of partitions of a finite set 
1d

revised 
Coarsest admissible topology on $\text{Cont}(X,Y)$
fixed a link 
1d

revised 
What technical and/or theoretical challenges are involved in automatically extracting proofs from books and papers into Coq code?
corrected spelling 
1d

comment 
Coarsest admissible topology on $\text{Cont}(X,Y)$
@EricWofsey I'm afraid it's not immediately obvious to me. But in the key case where the base $Y$ is Sierpinski space, see section 4 of the paper by Escardó and Heckmann where they show that the Scott topology is the exponential topology. 
1d

revised 
Coarsest admissible topology on $\text{Cont}(X,Y)$
special > general in "adjoint functor theorem" 
1d

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Coarsest admissible topology on $\text{Cont}(X,Y)$
I was in the middle of constructing my answer before I saw your edit. 
1d

answered  Coarsest admissible topology on $\text{Cont}(X,Y)$ 
1d

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What technical and/or theoretical challenges are involved in automatically extracting proofs from books and papers into Coq code?
In view of broadness (as mentioned by Theo), let me make this Community Wiki. 
1d

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What technical and/or theoretical challenges are involved in automatically extracting proofs from books and papers into Coq code?
You say "extracting ... to" instead of "extracting ... from", so that it's not clear to me whether you're more interested in extracting human proofs from Coq code or extracting Coq code verifications from human proofs. I'm guessing the former (?). 
2d

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To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz
Crossposting (asking the same question on multiple sites) is frowned upon, because it can lead to duplication of effort and hence a waste of people's time. You should wait for an answer at M.SE before giving up and trying here. 
Jul 2 
comment 
The problem of finding the first digit in Graham's number
There's a 70% chance it's not $1$. 
Jul 1 
comment 
What are a couple of examples of finite sized but interesting categories?
Well, the category of finite preorders is not a finite category, so I'm not sure what you're really after. The Monster group is an interesting finite category with one object and quite a few morphisms. The poset of truth values $\{0 \leq 1\}$ has a lot of nice properties (e.g., being cartesian closed) that are worth contemplating. Something that you might wish to contemplate is why the category of finite categories does not have coequalizers! 
Jun 29 
comment 
Fields of mathematics that were dormant for a long time until someone revitalized them
Gee, I'd add the general theory of buildings to that list. 
Jun 29 
comment 
tessellation of an arbitrary shape
Crosspost: math.stackexchange.com/questions/1343867/… 
Jun 29 
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Ways to prove the fundamental theorem of algebra
@DavidRoberts I really don't know. I'm having some difficulty finding the reverse mathematics of FTA through a Google search. Interesting question, though. 
Jun 29 
comment 
Good books on Geometric Theory of Dynamical Systems
As it stands, this answer is too vague to be useful. If there are books in that list that you can vouch for as being good for the OP's purpose, then you should say what they are (and why). 
Jun 29 
comment 
How to write an abstract for a math paper?
@DavidRoberts You didn't see me plagiarize! 
Jun 28 
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How to write an abstract for a math paper?
Bozhe moi! (and yes, this is a good answer) 