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May
2
comment Bayesian statistics for pure mathematicians
Tom, did you end up finding a book to your liking? I too wish to understand the Bayesian framework from a rigorous, pure math standpoint, so I'd appreciate any suggestions or advice.
Apr
28
comment Why forgetful functors usually have LEFT adjoint?
I strongly object to the notion that a forgetful functor ought to be faithful, and I also strongly object to the notion that a forgetful functor ought to have a left adjoint...
Apr
25
comment Seeking more information regarding the “rigoidal category” of $\mathbb{N}$-graded sets
Qiaochu, you write that: "the category of finite sets and bijections is the free symmetric monoidal category on a point." What happens if we replace "symmetric monoidal category" with "symmetric monoidal category equipped with an endofunctor $X \mapsto -X$ and natural isomorphisms $X \oplus -X \cong 0, -X \oplus X \cong 0.$" Do we get a categorification of the integers?
Apr
21
comment Why are matrices ubiquitous but hypermatrices rare?
"All aspects of matrix operations that I know (multiplication, determinant, etc.) have direct generalizations to tensors." How do you propose to define multiplication of tensors in such a way as to generalize multiplication of matrices?
Apr
20
comment Is injectivity of $2^{(\ldots)}$ weaker than $\mathsf{GCH}$?
You may find this relevant.
Apr
12
comment Is there a theory of abuse of notation?
Related.
Apr
4
comment Examples of math hoaxes/interesting jokes published on April Fool's day?
oh. This just seems like a "mere" hoax, without very much in the way of humour (to me at least). It would be better if it were more explicitly tongue-in-cheek, I think.
Apr
4
comment Examples of math hoaxes/interesting jokes published on April Fool's day?
I don't get it; what was the joke?
Mar
14
accepted Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$ and the usual notion of multiplication?
Mar
6
comment Does the Axiom of Choice (or any other “optional” set theory axiom) have real-world consequences?
@GeraldEdgar, what is the name of this essay?
Mar
5
comment Do names given to math concepts have a role in common mistakes by students?
I didn't even realize this was a thing! For me, the word "correspondence" just means "binary relation between two sets."
Mar
4
comment Can we strengthen the axiom of choice to settle the generalized continuum problem?
@ToddTrimble, to give a little more: it might be possible to prove GCH using a strengthened axiom of choice AC+ as follows: first, assign to each cardinal $\kappa$ a special relation $R_\kappa : 2^\kappa \nrightarrow \kappa^+$. Then prove that $R_\kappa$ is "sufficiently dispersive" that some kind of AC+ axiom implies that it has a subinjection. Hence conclude that $2^\kappa = \kappa^+$. Personally, however, I'm more interested in strengthenings of AC that imply the existence of at least one cardinal $\kappa$ satisfying $2^\kappa \neq \kappa^+.$
Mar
4
comment Does the Axiom of Choice (or any other “optional” set theory axiom) have real-world consequences?
For a dumb example: X could be {ZF + "If AC is true, then ZFC is consistent."}
Mar
4
comment Does the Axiom of Choice (or any other “optional” set theory axiom) have real-world consequences?
Sorry, my previous comment was typo-ridden. What I meant was: we can potentially "test" AC by considering the $\Pi_1$ consequences of X+AC where X is a collection of axioms distinct from ZF.
Mar
2
comment Can we strengthen the axiom of choice to settle the generalized continuum problem?
@JoelDavidHamkins, well, its not supposed to refute the existence of large cardinals, so consistency would certainly be a good first step in that direction :)
Mar
2
comment Can we strengthen the axiom of choice to settle the generalized continuum problem?
@ToddTrimble, to answer your second question; I have no particular reason to think that a strengthening of AC could settle the generalized continuum problem, except for the fact that GCH implies AC, and the feeling emanating from Hall's marriage theorem that there might be more powerful choice principles, and the feeling that combinatorial axioms (like a strengthened axiom of choice) should be "more powerful" than axioms like GCH which merely assert something about the structure of a decategorified $\mathbf{Set}$.
Mar
2
revised Can we strengthen the axiom of choice to settle the generalized continuum problem?
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Mar
1
comment Can we strengthen the axiom of choice to settle the generalized continuum problem?
@AsafKaragila, for the purposes of this question, the problem with that version of AC is that it has no potential to be strengthened by weakening the conditions on $R$, because there are no conditions on $R$, because all we're asking for is a subfunction. That's why I'm using the subinjection formulation here.
Mar
1
revised Can we strengthen the axiom of choice to settle the generalized continuum problem?
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Mar
1
comment Can we strengthen the axiom of choice to settle the generalized continuum problem?
@ToddTrimble, yep; its a made-up term, but seems self-explanatory enough that I thought it was safe not to define it. I also like the term "subfunction." One version of the axiom of choice says: "every total relation has a subfunction."