# Compatibility of Łośian phenomena in second-order logic

(Throughout, all ultrafilters are nonprincipal.)

Given a property $$P$$ - really, a sentence in some appropriate logic - say that a ultrafilter $$\mathcal{U}$$ on a cardinal $$\kappa$$ averages $$P$$ iff for every $$\kappa$$-tuple of structures, the ultraproduct by $$\mathcal{U}$$ of that tuple has $$P$$ iff $$\mathcal{U}$$-many of the structures in the tuple have $$P$$. For example, Łoś's theorem says that all ultrafilters average all first-order-expressible properties. Meanwhile, non-first-order properties are "usually" not averaged by ultrafilters.

However, even looking at a single strong logic we may see genuinely different "levels of averageing" amongst ultrafilters: e.g. looking at $$\mathsf{SOL}$$, note that well-foundedness is averaged by exactly the countably complete ultrafilters. This motivates the separate notion of "averageability:" a property $$P$$ is averageable iff there is some ultrafilter averaging it, and a set $$\Gamma$$ of properties is averageable iff there is some single ultrafilter which averages each property in $$\Gamma$$.

Interestingly, I don't know offhand of any examples of divergent behavior in any natural logic. To keep things precise, I'll focus on $$\mathsf{SOL}$$:

Question 1: Is it consistent with $$\mathsf{ZFC}$$ + large cardinals that the union of two finite averageable sets of second-order sentences is again averageable?

Now question $$1$$ is of course extremely broad. We can try to sharpen it by restricting attention to a particular type of ultrafilter. There are, to my mind, two natural ways to do this:

Question 2: What happens if we restrict attention to nonprincipal ultrafilters on $$\omega$$?

Question 3: What happens if we restrict attention to nonprincipal countably closed ultrafilters (and assume that lots of measurable cardinals exist, to avoid the trivial answer)?

Personally I think that a positive answer to $$(1)$$ is highly implausible, but a positive answer to $$(3)$$ might follow from (very) large cardinals. $$(2)$$ is pretty mysterious to me; large cardinals don't really tame ultrafilters on $$\omega$$ as far as I know (nor does anything else to be fair), so I'm not really sure where to get started intuition-wise.

I'm not sure I have a definitive answer, but three nice observations that are too long for comments:

1. If you relax from ultrafilters to extenders (seen as directed systems of ultrafilters), then there's a much stronger connection with large cardinals. In particular, the moral of Magidor's proof that extendibles are compactness cardinals for $$\mathbb{L}^2$$ is that the embeddings $$V_\alpha \to V_\beta$$ are $$\mathbb{L}^2$$-correct for structures in $$V_\alpha$$. So then all classes are averageable-by-extenders (although maybe this is slightly off from your definition because you need different extenders based on the rank of the models...)

2. Say that $$\phi \in \mathbb{L}^2$$ is existential if all of the second-order quantifiers are existential (and there's no negation outside of them). Then it is the case that existential $$\mathbb{L}^2$$ properties are averageable upwards, that is, if $$U$$-many $$M_i \vDash \phi$$, then $$\prod M_i/U \vDash \phi$$. So if you say $$\phi$$ is $$\Delta_1$$ iff $$\phi$$ and $$\neg \phi$$ are existential, then the $$\Delta_1$$ properties should exactly be the universally averageable' properties.

3. Beyond the $$\Delta_1$$ statements, I don't think there's going to be too much that is averageable within the realm of second-order logic specifically. The example you mention of well-foundedness is really a phenomena where well-roundedness is $$\mathbb{L}^2$$-definable AND $$\mathbb{L}_{\omega_1, \omega_1}$$-definable, and this second one is what allows its preservation by countably complete ultrafilters.

To back up this intuition, one way to think about this is to imagine all of the $$M_i$$ as Henkin structures $$(M_i, \mathcal{P}(M_i))$$. Then a second-order statement is just first-order in this structure, and so is preserved in moving to the ultra product $$\prod (M_i, \mathcal{P}(M_i))/U$$. But there is a big gap between

• $$\prod \mathcal{P}(M_i)/U$$, the subset sort of the Henkin structure and
• $$\mathcal{P}(\prod M_i/U)$$, the actual power set of the ultraproduct that the second order statements will check

This is why universal second-order statements are not averageable up', because they miss the extra subsets in $$\mathcal{P}(\prod M_i/U) - \prod \mathcal{P}(M_i)/U$$

I guess a question to help guide any of this intuition is the following: do you have any candidate properties in mind that are averageable but not a) uniformly averageable (by dint of being $$\Delta_1$$) or b) equivalent to some $$\mathbb{L}_{\kappa,\kappa}$$-property and are averageable only for $$\kappa$$-complete ultrafilters?

• Nice observations, Bill. Nov 17, 2021 at 21:53
• Re: (2)/(3) and the end, I don't really have any such interesting candidates but I'll see if I can come up with one. (Maybe something from a fully-compact logic beyond FOL?) Re: (1), I'm happy to generalize to extenders but I really do care about not depending on the input structures, so I'm not fully satisfied with that. Nov 18, 2021 at 4:33
• An addendum to the $\Delta_1$ observation above: a better way of looking at it is that existential second-order sentences are preserved by ultraproducts is that the are PC-classes and reduct commutes with ultraproducts. Nov 18, 2021 at 22:33
• One more addendum: I think averageability of some honest/unavoidable use of a universal quantifier for all structures is going to be very hard. As M gets big, the different between $P(\prod M)$ and $\prod P(M)$ is going to get bigger and bigger, so you'll keep missing out on more subsets that you need to universally quantify over. This is why I think an example would be very instructive, since you'll start to see where averageability fails. Nov 19, 2021 at 17:59
• This is quickly getting out of my realm of competence, but this answer of Farmer S seems relevant. (Today's regret: why didn't I pay more attention in John's seminars?) Dec 24, 2021 at 19:42