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 Mar18 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. Mar18 comment How to define a truth set that is the entire domain in logic? Well, there's always $\{x\in\mathbb{Z}:x=x\}$. Mar17 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 Mar17 accepted Can a general version of the covariant powerset monad be derived from the universal property of power objects? Mar16 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 Mar16 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. Mar16 asked Can a general version of the covariant powerset monad be derived from the universal property of power objects? Feb14 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 $\alpha_{m>n}$. Jan8 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. Dec30 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. Dec28 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. Nov28 awarded Necromancer Oct23 awarded Yearling Sep30 awarded Explainer Sep27 comment Are any two uncountable sets similar to each other? @VHP: Is $A \subseteq B$ in your problem? Sep27 comment Are any two uncountable sets similar to each other? Well, any two countable sets are similar because "countable" refers to only one cardinality, and similarity is the definition of being equinumerous. But Cantor's theorem shows you that there's always a properly bigger set if you can form the powerset. Sep25 comment What is an axiom schema? It's usually a set characterized syntactically. They're all alike modulo some relettering of variables and plugging in subformulae in the right spots. Sep17 comment Why can't a set have two elements of the same value? You're on a road trip with three friends, Eiko, Biko, and Shiko. You decide is so much fun that next year you want to go on a trip with Eiko, Eiko, Biko, and Shiko to change things up. Except you don't decide that, because the idea is absurd. Sep14 comment Can the logic associative law be applied here? Well that's the thing here: associativity isn't part of the definition of implication, so if it holds you have to prove it. Sep12 comment Adjoint functors for the power set monad Yes, though I've seen it referred to simply as the "powerset monad". It sounds like there are others I'm not familiar with, though.