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Wrote cumulative hierarchy, and edited some at ZFC (idea section, reference, related articles).
Should this be connected to algebraic set theory? This is really an algebraic construction.
Oh sure, I had meant to say something there but forgot; please feel free to jump in yourself (and thanks for the reminder).
It was only recently that I realised how much set theorists like the cumulative hierarchy, and this is one reason they (or at least some) find structural sets less than satisfying: it’s not obvious that, say, ETCS comes with anything like being able to say “where the sets came from”. I guess in AST it’s different? One probably needs a form of Replacement to construct even for an arbitrary individual set a ’membership tree’ in the structural style. For sets with injection to some P^n(N) there is no problem, I think.
I have tried to say a few things about AST in cumulative hierarchy. This however is all from memory, as I am having difficulty tracking down my copy of Joyal-Moerdijk.
David, I remember a recent MO question of yours in which I started a long comment thread on the meaning of “ontology” for ZF-ists, and it seems that yes, it’s all about the cumulative hierarchy – although I don’t recall anyone actually using that phrase! (For me the most helpful replies were from François Dorais.) There are still a few mysteries for me from that thread, which I may get back to at some point.
I see that Thomas Holder has helpfully added a related articles section; however, some of those nLab articles could use a little more love. :-)
One probably needs a form of Replacement to construct even for an arbitrary individual set a ’membership tree’ in the structural style.
What do you mean? Do you mean under what conditions is an ETCS-set isomorphic to the set of nodes of a rigid wellfounded tree (or whatever)? I think the answer to that question is “always”, because ETCS includes AC so that every set can be well-ordered and is hence isomorphic to a von Neumann ordinal.
@Mike whoops! I must have been thinking of more general toposes as models for set theory… (I’d looked at the relevant section of the Baby Elephant recently) :-S
@Todd - yes, it was that discussion that made me realise this. The fact that ZFC sets are exactly subsets of some set constructed so far is still far more rigid than the case for a structural approach: one merely has a monomorphism $X \hookrightarrow P^\alpha(\mathbb{N})$. So I guess a material set theorist would ask: “where did the domain of that mono come from?” But working up in an isomorphism-invariant way we just have to bear with that, as it were. Perhaps thinking of monos as elements in the powerset $P^{\alpha+1}(\mathbb{N})$ (say we’ve chosen an initial object $1$ once and for all: one choice instead of many) we get a bit tighter…
Also: there’s the cumulative hierarchy constructed in the HoTT Book.
@Mike actually, I don’t think I do mean that. Yes, one gets an isomorphism to the nodes of a rigid wellfounded tree from a well-ordering, but there is a massive amount of freedom. In particular, one might as well always choose an initial ordinal. What I’m thinking of is whether given any set $X$, one can show the existence of some ordinal $\alpha$ (with no choices, or at least none explicitly or implicitly well-ordering $X$) such that $X \leq P^\alpha(\mathbb{N})$. Such a result would not rely on Choice in the same way. Whether only well-orderable $X$, in the absence of choice, admit such an embedding (it obviously doesn’t a priori well-order $X$), is an interesting (to me at least) problem.
Compare the problem of constructing something like $L$ inside a topos: its definition in material set theory doesn’t require choice.
Cf also A cumulative hierarchy of sets for constructive set theory
@David, do you just want to work in ETCS-minus-AC? I wouldn’t think $X\le P^\alpha(\mathbb{N})$ would imply well-orderability of $X$; doesn’t it follow in ZF from the axiom of foundation?
@Mike ETCS-AC is as good as any to start thinking about such a problem.
I wouldn’t think $X\le P^\alpha(\mathbb{N})$ would imply well-orderability of $X$
No, I didn’t think so. The trick is finding an ordinal such that it works. And, of course, one may not be able to use $\alpha \geq \omega$, since $P^\omega(\mathbb{N})$ may not exist in the topos: some Replacement may be needed.
You’re probably right about Foundation, it sounds familiar.
And all this is only at the level of musing, I’m not actively pursuing a result.
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