good question; is anything here useful to you? ncatlab.org/nlab/sh…

huh! This is apparently just about exactly what I’ve been thinking about here, in fact a lot of the theory developed in these past few posts appears in those pages, which is neat :V (they go a bit further and see things a bit more clearly, though, I think.)

It’s neat, it looks like the field isn’t super explored though, going by the number of citations and the amount of detail on the “material set theory” and “structural set theory” pages (not a lot).

huh

guess it was interesting after all, who knew

I’d like to point out one thing though: this is proposing constructing and adjoint functor between the category of material set theories and the category of structural set theories. In other words, this is proposing constructing a functor between the Category of Categories of Sets and the Category of Categories of Material Sets.

just uh. so that the foundational issues are obvious.

now that I’ve said this, I’d also like to suggest that an interesting problem might be to extend this adjunction outwards so that we have, idk, elementary toposes? on the Category of Categories of Sets side? basically I’m just curious if we can get a membership-using set theory that corresponds to, idk, the category of sheaves on whatever. just for fun, yknow…

(this would probably involve making it a fair bit more precise, to be clear on the challenge. if it’s not possible with axioms that weak though, one might just add some until it exists or whatever…)

thesummerofmark replied to your post: apparently my current mood is “designi…

gonna sell ‘em on Etsy?

i was considering switching the colors to black+red and putting some of these on Print All Over Me’s work shirts, so you can look like some kind of lumberjack from another dimension, but that would cost me 15\$/month (or 100\$/year) so I would want to know that at least one person would be willing to buy one (they cost 85\$, for 17\$ profit for me)

(that or their skirts)

(or some number of other products)

Oh, ok, it was just a notation problem ^^. Anyway, it's a pity that your asertion about that cathegory and the union and intersection is not true, I really wanted to try to understand it. I thought that, f it had been true, it would have been a very beautiful result!

Well, one can still have some fun figuring out what, exactly, the product and coproduct are in those categories, I think. I think it might be tricky to write down, but it should be doable.

I’m still kind of bothered by this. Like, as far as I can tell, well-foundedness is something that isn’t captured at all by the category theory of the category of sets: as far as functions are concerned, what determines a set up to isomorphism is cardinality, and there are no new cardinalities added by adding non-wellfounded sets, thus, the categories of well-founded and non-well-founded sets are equivalent (uh, there’s a different, more precise argument to be made, but it’s not very hard tbh). So, where does all of this stuff live, from the perspective of category theory? Where does the set theory that isn’t captured by the category of sets live?

And I mean, also, I guess this might capture an idea that the notion of elements themselves having elements isn’t really terribly natural in a lot of mathematical contexts, but it’s totally necessary in others, and, idk, I’m just kind of curious about whether the information lost when considering sets as objects the category of Sets can be recaptured otherwise, just so that I can know what was lost and characterize when it’s useful.

or something

almost all the math i’ve ever written down has been wrong

To be clear, eventually I figured out that what I was looking for (the category whose products and coproducts are intersections and unions) is just, the category where the maps are set maps that preserve all set structure, in other words set maps f such that f(x)=x, in other words, inclusions, in other words, this is just the Poset of sets ordered by inclusion.

In other words, it was boring.

I've got two questions about your posts on the ZFC sets seen as digraphs. How do you define an "open neighborhood" in a digraph? and What do you mean by "top element"? Sorry for ask these questions, which might be trivial ^^

The “open neighborhood” of a vertex x in a digraph G is the set of vertices y in G such that xy is an edge in G. In this case, the digraph is also an “accessible, pointed” digraph, which means that there’s a node in it that is denoted the “root” and every other node should be accessible from that node. This nodes corresponds to the set itself, and the arrows represent “having as an element”.

Anyways, the point is I was thinking of that node as the “top” node, and the arrows as going “down” to lower sets, because the arrow essentially always goes from a bigger to a smaller set (although this does go the wrong way relative to the “root” analogy).

(Actually my talking about these like that is probably the worse possible notation, since the condition that no two nodes can have the same open neighborhood means that there can be only one node with no edges going outwards from it, which means that there is exactly one node that is “furthest” from the root, which means that in fact the graph some uniquely defined “top” (away from the root (it’s the empty set)) and “bottom” (the root itself (it’s the entire set)) :V (but whatever))

((I’d also like to note that the thing I said about the product and coproduct operations in the category that preserves digraph-structure, top, and bottom being intersection and union is, in fact, wrong. oh well.))

Oh, so when you say you spent years dealing with and eventually learning how to communicate with people like me with a lot of effort, you’re either “an inspiring story of acceptance” or “a medical professional”, but when I say I did the same thing with you guys I’m “exaggerating” and “not actually autistic”.

Tbh the main reason i kinda like skype better than a lot of other chat things is that in skype, you can edit previously written messages. And that makes it possible for me to communicate in a way that is somewhat closer the way I think.

actually yeah that definitely works

if you take the category of ZFC digraphs (as defined previously) and then say that morphisms must preserve digraph structure + top element + empty set element, then product is the actual usual set-theoretical intersection and coproduct is the actual usual set-theoretical union.

sort of annoying though, I was hoping I’d get something that generalized ordinal sum imo…

oh well, w/e

Y’know what that means though. Given all pointed directed graphs, what you need to specify ZFC is

1. Accessibility
2. No infinite paths (wellfoundedness)
3. All points have disjointdistinct open neighborhoods (extensionality)

All the other axioms are hidden in the word “all” up there…

It doesn’t have to be disjoint. \( \{\varnothing , \{\varnothing \}\} \) is the graph below. The red and green points’ neighborhoods both contain the empty set. They have distinct open neighborhoods, because the green point is not in its own neighborhood. In fact, well-foundedness is equivalent to “No point is in its own open neighborhood.”, which is a better formulation since cycles are not normally considered infinite.

I said disjoint, I meant distinct. This is why I should eat before I post. oops.