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Jun
29
comment How do I find the type of relation on an infinite set?
(Actually I misstated things in my above comment; the second term of the second pair needn't be $\{\{a\}\}$, but it will still turn out that $a$ will need to be a member of every $x\in\mathcal{P}(A)$ such that $\{a\}\in x$, which leads to the same problem.)
Jun
29
comment How do I find the type of relation on an infinite set?
For transitivity, note that the question only arises when you have pairs of the form $(a,\{a\})$ and $(\{a\},\{\{a\}\})$. For the relation to be transitive it needs to be the case that $a\in\{\{a\}\}$ for all $a$. (The general methodological note here is not to worry about the picture of a list of stuff in curly braces, but to go back to the logical conditions that defined your sets.)
Jun
26
comment Do comonads always induce cosimplicial objects? Vice-Versa?
What one specifically gets from a comonad $L$ on $\mathcal{C}$ is a functor $F:\mathcal{C}\to \mathcal{C}^{\boldsymbol{\Delta}^{op}}$. So I doubt one could generally take a single simplicial object and get a comonad back out of it since, as you suppose, that's a lot of information to ask from one simplicial object. I have no idea about obtaining a comonad from, say, some sufficiently nice subcategory of $\mathcal{C}^{\boldsymbol{\Delta}^{op}}$, but I hope someone has an answer to that!
Jun
24
comment Why composition is so important in category theory?
Not only does it abstract procedural idioms as Sr. Suárez-Alvarez mentions, but it naturally falls out of the behavior of the quantifiers of the underlying logic; if every person has a right hand, and every right hand has a thumb, then every person has a right thumb. It's an artifact of being able to instantiate universal quantifiers as arbitrary values.
Jun
8
comment Non WellFounded Set theories and Russell's Paradox
Moreover, there's an easy construction that can take a model of ZF and give you a model of ZF-foundation+"there is a Quine atom", showing that the latter is consistent if the former is.
Jun
4
comment How to think of a set?
@creator: Well, it's not just the informal notion of a collection that allows us to implement mathematical objects, but the properties we suppose the universe of sets to have in the formal theory of sets. It's an important distinction to make, lest your thinking get hung up on what you can do with some single collection (which isn't much).
May
14
comment How do you evaluate $\bigcup\bigcup S$ if $S = \{\{a\},\{a,b\}\}$?
The single union of $\{\{a\},\{a,b\}\}$ certainly is $\{a,b\}$, so why would taking the union a second time give you $S$ again?
May
6
comment Is there a “properly categorical” description of Eilenberg-Moore algebras on a relative monad?
I'll look into those! I'm generally comfortable with the category-ness of relative monads, since in essence all it says is that the initial splitting of $F$ into a $J$-relative adjunction exists; and relative adjunctions make sense in nice 2-categorical terms. It's when it comes to the EM algebras that I find even the picture of a composite of relative adjoints unhelpful.
May
5
comment Limiting the set of “constructible” properties, and loosening comprehension axiom
There is also positive set theory, regarding which Olivier Esser has written many good papers.
May
4
asked Is there a “properly categorical” description of Eilenberg-Moore algebras on a relative monad?
May
4
comment Does type-theory have a concept of “relation”?
The second quoted instance is one of the more general formalizations of relations in type theory that I'm aware of. One might like $R(a,b)$ to have nicer properties than an arbitrary type (like being contractible or having anonymous proofs), but a relation between elements of types $A,B$ can be thought of as an element of $\prod_{a:A,b:B}R(a,b)$.
Mar
18
comment How to define a truth set that is the entire domain in logic?
@Carl: There's an element of hyperbole in my comment to be sure, but I find it to be warranted as FOL with equality is more the rule than the exception.
Mar
18
comment How to define a truth set that is the entire domain in logic?
Well, there's always $\{x\in\mathbb{Z}:x=x\}$.
Mar
17
comment Can a general version of the covariant powerset monad be derived from the universal property of power objects?
Thanks! I'm glad I was wrong about this one. And this is a good reminder to me to use the bloody internal language... :P
Mar
17
accepted Can a general version of the covariant powerset monad be derived from the universal property of power objects?
Mar
16
revised Can a general version of the covariant powerset monad be derived from the universal property of power objects?
clarifying the parameters of the question
Mar
16
comment Can a general version of the covariant powerset monad be derived from the universal property of power objects?
I'm probably being dense, but how might one get the monad structure from that? I understand how one gets the functor part to work, but I've been stumped trying to show that the obvious candidates for unit and multiplication work. Maybe I should edit the question to make that a little clearer.
Mar
16
asked Can a general version of the covariant powerset monad be derived from the universal property of power objects?
Feb
14
comment Does every countable sequence of ordinals converge?
@user83081: If the $\alpha_n$'s are strictly increasing that sort of takes care of itself, right? The union is an ordinal that has as one of its members $\alpha_n$, since this will itself be a member of $\alpha_{m>n}$.
Jan
8
comment Simple (even toy) examples for uses of Ordinals?
There are a number of decent, if unexciting, examples out of logic. The neatest way to describe Henkin constructions, or even just the extension of a consistent theory to a complete theory, is via recursion over an appropriate segment of the ordinals. They don't ask very much of the ordinals, but if one can explain why uncountable languages come up it's easy to see why a llonger ordering than $\omega$ is handy.