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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?
Sep
27
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.
Sep
25
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.
Sep
17
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.
Sep
14
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.
Sep
12
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.
Sep
11
comment Adjoint functors for the power set monad
In the powerset monad that I'm familiar with, the contravariant powerset functor is both the $F$ and the $G$, and the counit is the unit in $\mathbf{Set^{op}}$.
Sep
11
comment Can a function be applied to itself?
Since I always have to chime in with things like this, in NF(U) it's easy to construct examples of such functions (NF violates both foundation and Cantor's theorem). Let $g$ be any permutation of $V$, and let $Q$ be $\{\langle f,g\circ f\rangle : f\in Funct\}$. $Q$ is a function itself, hence it's in $Funct$, and therefore a member of its own domain.
Sep
10
comment quantification domain of set theory formulas
Why should it need to?
Sep
3
comment Recommendation on Category theory textbook
I really like ACC, though it should be mentioned it doesn't have quite the breadth of Mac Lane. Though it also covers some concepts I haven't seen in other book-length works.
Aug
30
answered Are $1+ω$ and $ω+1$ isomorphic?
Aug
30
comment Are $1+ω$ and $ω+1$ isomorphic?
Yes, I'm quite familiar with ordinal arithmetic. That wasn't the part that made your comment opaque. Though if you're talking about concatenation of wellorders you may not want to write it as an ordinary union.
Aug
30
comment Are $1+ω$ and $ω+1$ isomorphic?
"Inputed"? (Not trying to be pedantic, actually trying to figure out your comment.)
Aug
30
comment Are $1+ω$ and $ω+1$ isomorphic?
$\mathbb{N}\cup\{a\}$, without further information, isn't necessarily wellordered. One needs to know what happens to the relation $\leq$ when $a$ is added.