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Alice Vidrine
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Alice Vidrine

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Dear mathemativerse,

Are "right exact functors" a thing? And are they, as one hopes, the dual of left exact functors?

Not enough of a question to go on Stackexchange, but enough that I had to put it out there.
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Alice Vidrine

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Because I feel like throwing down...

Apparently there's now a name for the constructivist confusion between alethic logic and a theory of proof: "proof relevance." Because clearly anything that can be given the clothing of formal logic, is logic.

Okay, now I'm done being curmudgeonly.
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Alice Vidrine

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Whoa, wait, what? Olivier Esser's positive set theory can interpret MK?
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Alice Vidrine

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I seem to be dead set on creating a theory of ordinals for ML just so I can go on and do model theory in it. Why not just keep learning model theory in a ZF-like theory and worry about this later? Because it's insufficiently beautiful and that is NOT HOW I DO.
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Sounds like I was in the right neighborhood, then! You know, minus the hand-waving and such.
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Okay, ML is just fucking with me again. I just found an example of three functions, each apparently proper classes, which, when composed, form a function which is a set. Maybe they aren't all proper classes?

That's the painful thing about ML; all of the classes involved in the above thingamy are the same size, but of the five classes involved (the three functions and the two sets they act on) only one of them is a set. Sometimes I wonder why I don't just settle for MK...
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There's already the clear example of the singleton function between V and 1, but I found this one more surprising. I think the composite was the successor function?
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Alice Vidrine

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Aaaaand.... UNCLICK I fucking hate model theory.
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Alice Vidrine

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I would like to take a time out to give a nod to Alfred North Whitehead. His mereotopology was the first theory I ever studied using formal logic, rediscovering their inconsistency. Relatedly, a nod to Casati & Varzi for their book on various systems of mereotopology, particularly GEMTC, which helped motivate, oddly, my first foray into category theory (by way of models of GEMTC in a topos). For a seldom used theory of connected parts, it turned out to be a pretty deep rabbit hole for me.

(Ran across some old proofs in a notebook.)
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Alice Vidrine

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My recent notes are pointing to a very intimate relationship between the non-Cantorian parts of NF, the failure of Cartesian closedness, and the failure of AC. They're all really entangled, and this entanglement only kind of depends on the fact that for some sets, NF's internal X^1 is not isomorphic to X. I'm still hoping to find some more positive aspect of this entanglement; probably some characterization of what NF's "exponentials" really do that leads to the automatic failure of real exponentiation and the Pi functors, which in my wild fancies I imagine will give us an account of what the "USC" functor does in categorical terms...

But for now, the concrete problems...
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I do think that NF itself might have specific properties which a category theoretic approach will capture which are not characteristics of NFU in general.  But I dont think in category theory.
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Alice Vidrine

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Hm, I thought with type-level exponentiation that I'd be able to easily define, and prove the existence of, very large Beth numbers. It turns out I really don't get how to talk about cardinals in NF+Counting/ML+N.

For one, I don't even know how I'd define beth_omega, much less prove (as I would need to) that it is Tc for some cardinal c. I mean, outside of invoking it as a supremum that might not exist in these set theories. Forster explained part of the solution to this, but not enough that I can piece the details together.

I dislike feeling like I need my hand held on things like this.
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notice this works just as well in ZF without choice and the motivation is
more obvious.


On Thu, Oct 17, 2013 at 5:56 PM, Randall Holmes
<****@**>wrote:

> Its a general solution in basically any set theory. Choice is not needed
> at all to describe the beth numbers.
>
> The cardinality of the set of all isomorphism types of [well-founded
> extensional relations with top element] of rank alpha in an obvious sense
> is beth_alpha.
>
> The reason is that this class of relations is precisely the class of
> isomorphism types of restrictions of the membership relation to elements of
> V_alpha.
>
> Of course that motivation is obscured in NF, but the definition works just
> as well, and is quite independent of choice.
>
>
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Alice Vidrine

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Jeez, no wonder the sethood of the natural numbers proves the consistency of ML. Can we say Beth numbers at least up to omega*omega?
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No worries! To be fair, I only clued in on half this stuff starting to talk to Forster and you, so some of this stuff was still current until very recently.
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A ramble about the class of well-orderings in ML.
Huh, I'm finally seeing where the original form of ML ran afoul of Burali-Forti, and it's making me feel like ML is actually diminished in many respects compared to NF. NF has a set of well-orderings, and that is a subset of ...
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Alice Vidrine

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So I know a lot of what I post here comes across as patting myself on the back for being kind of smart. Well that's what I'm doing explicitly for this post.

I got curious as to what it would mean for half of the fundamental theorem of topoi to hold for a general set theory, specifically to see if its definition would work in NF. Using only very basic properties of Set I figured out what would work as the right adjoint to a pullback functor. It's basically a family of sections of restrictions of a morphism.

I have a suspicion, though I need to do the actual checking, that the existence of such a right adjoint for any pullback functor relies on something like the axiom of choice, since it relies on there being arbitrary sections. I'm curious how strong a version of choice it entails, if I'm correct.
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  • University of Washington
    Psychology, 2000 - 2005
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This time, there can be only one danger, for love.
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I am made partly of ink.
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I am pretty good at poi, and have taught myself formal logic and basic group theory and category theory. I also invented marmots.
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