(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' 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.

(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.

(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, Łos' 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.

(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' 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.

(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, Los' 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.

(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, Łos' 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.