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• CommentRowNumber1.
• CommentAuthorTobyBartels
• CommentTimeOct 31st 2019

Add Mike's stack semantics paper to the references (for its classification of axioms of material set theories).

• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeOct 31st 2019

Have I forgotten how to link an internal bibliographic reference, or does that no longer work?

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeOct 31st 2019

Actually, maybe the spin-off paper that contains just the material/structural set theory material would be better?

(OT: I’m hoping the spin-off stack semantics paper will turn up too, at some point)

• CommentRowNumber4.
• CommentAuthorAlexisHazell
• CommentTimeOct 31st 2019

@TobyBartels: re. #2, could you describe the specific issue you’re having?

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeNov 1st 2019

I’m hoping the spin-off stack semantics paper will turn up too, at some point

It will…

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeNov 1st 2019

Thanks :-) I’m looking at having a student do their honours project next year on the stack semantics (mostly a survey of toposes and the internal logic, with end goal of proving a particular statement in the internal logic holds in a particular example).

• CommentRowNumber7.
• CommentAuthorTobyBartels
• CommentTimeNov 1st 2019

@AlexisHazell #4 : I was indeed formatting it incorrectly; I referred to another page and fixed it. (The issue is that if you want to attach an id tag to an li element, then it has to go in a separate line that does not have the indented spacing that you would use to keep a new paragraph in the list item.)

• CommentRowNumber8.
• CommentAuthorTobyBartels
• CommentTimeNov 1st 2019

@David Roberts #3 : Yes, that is better. (I didn't realize that it was publicly available yet.) I've changed it.

• CommentRowNumber9.
• CommentAuthorDavidRoberts
• CommentTimeNov 1st 2019

Updated reference to Mike’s paper: 2018 not 2010.

• CommentRowNumber10.
• CommentAuthorDavidRoberts
• CommentTimeNov 1st 2019
• (edited Nov 1st 2019)

I think it might be worth saying something about $\mathrm{ZF}(\mathrm{C})-$, Zermelo–Fraenkel without the axiom of powerset, as it turns up now and then in various settings. But there’s subtlety in that other axioms need fiddling with (or adding, if they no longer follow from the rest of them). This paper takes the ’$\mathrm{C}$’ to mean the well-ordering principle, rather than AC. There also seems to be work of Andrzej Zarach in this area.

I guess it might also be worth also isolating the structural analogues of $\mathrm{ZF}(\mathrm{C})-$. For $\mathrm{BZ}-$ (whatever that might be) one probably gets something like a well-pointed Boolean pretopos with NNO and subobject classifier. I guess carefully adding one of the now non-equivalent stronger structural replacement-ish axioms gets us to $\mathrm{ZF}-$.

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeNov 2nd 2019

IMO in almost any setting where replacement no longer implies collection, in practice one wants to assume collection rather than replacement. I think that was also the lesson of the GHJ paper.

The paper of mine cited in #3 compiles a fairly extensive list of structural axioms that correspond to various axioms of ZFC.

• CommentRowNumber12.
• CommentAuthorDavidRoberts
• CommentTimeNov 2nd 2019

@Mike Ah, I didn’t see that the GHJ ZFC- was listed in your paper. I’m really looking at ZF-, so I will see if I can extract something. Krapf’s thesis (for example) gives ZF- as

1. extensionality,
2. foundation,
3. pairing,
4. union,
5. infinity,
6. collection

Given $\mathbf{V}$ that satisfies these and we have classical logic, I would guess that foundation+infinity implies $Set(\mathbf{V})$ has a parametrised NNO. Collection+infinity implies empty set and also the weak separation and replacement from the core axioms (an educated guess). So $Set(\mathbf{V})$ is a well-pointed boolean pretopos. Then the core axioms + collection for $\mathbf{V}$ give structural collection for $Set(\mathbf{V})$. There’s probably more that can be said, but I’ll leave it there for now and come back.

• CommentRowNumber13.
• CommentAuthorTobyBartels
• CommentTimeNov 5th 2019

Also mention Mike's paper in the part of the article that is about that subject (comparison to structural set theories).

• CommentRowNumber14.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 6th 2020