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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.
Dec
30
comment Universal object
In studying something like class theories the "every object injects into it" is pretty standard. See Colin McLarty's paper about Cartesian closedness failing in NF, or the definition of a universe in algebraic set theory. There may be finer notions that are handy in different contexts, though.
Dec
28
comment Is F(x,y)=x an atomic formula?
Predicate symbols with the appropriate terms form formulae, not terms. So if $P$ is a binary predicate symbol, $P(x,y)=z$ or $P(x=y,z)$ are gibberish. Identities between terms and predicates on terms are all the atomic formulae there are.
Nov
28
awarded  Necromancer
Oct
23
awarded  Yearling
Sep
30
awarded  Explainer
Sep
27
comment Are any two uncountable sets similar to each other?
@VHP: Is $A \subseteq B$ in your problem?