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Stream: learning: questions

Topic: reflection

Leopold Schlicht (Aug 29 2021 at 12:59):

In the paper https://arxiv.org/pdf/1304.5227.pdf, it says:
"more recently, Bowler [2012] has constructed an ω-sequence of monads on Set whose colimit depends on U, where Set is the category of U-sets."
Does this mean that he has constructed such an ω-sequence in any model of (Tarski--Grothendieck, say) set theory (is that possible?) or merely that he has constructed one model of (TG) set theory in which this happens?
It would be nice if there's a model of set theory in which such things as limits don't depend on the chosen universe. Or, better yet, nothing depends on the chosen universe. Is there a model of set theory satisfying the following very strong reflection principle?
"Let $\varphi(x_1, \dots, x_n)$ be a first-order formula and $\kappa$ a strongly inaccesible cardinal. Furthermore, let $x_1, \dots, x_n\in V_\kappa$. Then $\varphi(x_1,\dots,x_n)$ holds if and only if $\varphi(x_1, \dots, x_n)$ is true relative to $V_\kappa$ (i.e., when each quantifier in $\varphi$ is interpreted as a bounded quantifier over $V_\kappa$)."
If not, what is the strongest reflection principle which is known to have a model?

Morgan Rogers (he/him) (Aug 29 2021 at 15:43):

Were you unable to access the paper being referenced? I'm sure that would give you more insight than anyone here guessing!

Mike Shulman (Aug 29 2021 at 16:06):

@Morgan Rogers (he/him) That was my first reaction too, but if you follow the link you'll see that the citation is actually to a seminar talk. (-:

Morgan Rogers (he/him) (Aug 29 2021 at 16:26):

Aha, that makes sense. My apologies!

Leopold Schlicht (Aug 31 2021 at 20:30):

My hope was that the knowledgeable people here could tell me that the reflection principle I stated has a model (which also satisfies TG), which would exclude the first interpretation anyway. :P

Leopold Schlicht (Aug 31 2021 at 20:44):

I did a bit of reading and the closest I found is the theory ZMC/S from Mike's Set theory for category theory. But it seems TG + the reflection principle I stated is stronger.

Leopold Schlicht (Aug 31 2021 at 20:49):

A weaker reflection principle which would still be satisfactory is the existence of arbitrarily large Grothendieck universes reflecting all formulas.

Mike Shulman (Aug 31 2021 at 20:58):

It seems unlikely to me that it would be possible to model a reflection principle saying that all Grothendieck universes reflect the theory of the entire universe, which is what your original version seems to claim. The existence of arbitrarily large Grothendieck universes with this property seems more likely to me to be consistent relative to some large cardinal axiom. Just as ZMC/S is equiconsistent with "the universe is Mahlo", I would guess that maybe your reflection principle is equiconsistent with something like "there is a proper class of Mahlo cardinals" or perhaps "the universe is 1-Mahlo". But you'll probably get better answers to a question like that on a forum that includes more set theorists. I don't know if there is a set theory Zulip, but you could try MathOverflow.

Mike Shulman (Aug 31 2021 at 20:59):

You should probably also notify @Zhen Lin Low, who may be able to answer your question about the cited preprint.