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Created measurable cardinal.
Excellent. Thank you, Mike.
What references do you recommend for set theory? Are there any advanced structural accounts available?
No, I don’t think there’s much of anything written from a structural point of view. (Maybe one day we’ll change that.) You just have to read the materialist stuff and think “well-founded extensional relation” whenever they do something very $\in$-centric. (-: I don’t think I have any book recommendations that are different from what a material set theorist would tell you, like Kunen and Jech. I liked Kanamori for large cardinals specifically.
Well, that’s what I figured. :-) Actually, I’ve been thinking I should buy Jech.
Wrote the beginning of an article on ultrapower.
Kunen is a fantastic lecturer (unfortunately I attended just about 3 times one half a course each time from him).
I learned a nice thing about measurable cardinals, which I added to this page. There exists a measurable cardinal iff there’s an exact functor $F : Set \to Set$ that’s not naturally isomorphic to the identity!
I assume the idea is that if there’s a measurable cardinal, $Set$ is so big it becomes ’floppy’.
Anyway, I like the idea of finding category-theoretic equivalents of large cardinal axioms.
Yep, that’s due to Andreas Blass (as I see you’ve noted).
Here’s another. Let’s back up a minute, and consider the axiom of infinity as a large cardinal axiom! Now one such axiom of infinity is existence of a non-principal ultrafilter on a set (this implies the set is infinite, and under the axiom of choice we also have the converse direction).
As Tom Leinster had mentioned a while back at the Café, in one of his ultrafilter posts, an ultrafilter on a set $X$ is equivalent to a map $3^X \to 3$ between $Endo(3)$-acts (sets which carry actions of the monoid $M = Endo(3) = \hom(3, 3)$). It is principal if the map $3^X \to 3$ comes from evaluation at a point.
What we are dealing with here is an ambimorphic (or janusian or schizophrenic) adjunction, where we are viewing $3$ as living in two worlds, one world being $Set$, the other being $Set^M$. We have a contravariant adjunction induced by the ambimorphic object $3$, given by the two contravariant functors $Set(-, 3): Set \to Set^M$ and $Set^M(-, 3): Set^M \to Set$. We have a unit of the adjunction
$u_X: X \to Set^M(Set(X, 3), 3)$which maps $x \in X$ to the evaluation map $ev_x: Set(X, 3) \to 3$. The large cardinal axiom on $X$ here is that the unit $u_X$ is not an isomorphism.
One can play the same game with other ambimorphic adjunctions. You don’t get any new concept of ’large cardinal’ here if you replace $3$ by another finite set $n$. But:
Lawvere pointed out these facts in a post to Andrej Bauer at the categories mailing list. He mentions a few more examples of a more continuous (as opposed to discrete) character, which I don’t feel I understand properly, but inevitably of type where the existence of a unit map (of some ambimorphic adjunction of this monoid-action type) is not an isomorphism is equivalent to existence of a measurable cardinal. Something apparently along these lines is mentioned here, for instance.
My advisor, Joel W. Robbin proved in his early days (probably in late 1960s, unpublished) that the set-theoretic analogue (if I recall right that there is an antiequivalence of category of sets and of commutative unital algebras with involution by taking algebra of all complex functions in one direction and characters in another) of Gelfand-Neimark theorem (antiequivalence of category of locally compact Hausdorff spaces and of commutative $C^\ast$-algebras, in unital case just compact Hausdorff spaces) holds iff there are measurable cardinals. I think I have seen some refinements of this in the newer literature. I never thought it through though.
You’re making it sound like every commutative unital (complex) algebras with involution is isomorphic to one of the form $\mathbb{C}^X$ for some set $X$, at least if there’s a measurable cardinal. But how could that be? What about algebras with nilpotents, for example?
Anyway, some result of this general ilk would be very interesting to me - though not for any good reason, just because it sounds cool.
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