I meant that (conjecturally) for every measure, there exists a cardinal kappa such that mu({M: |M| > kappa}) = 0. Anyway, I guess as you’ve demonstrated the set/class thing isn’t a big problem, but it is something to watch out for.
Okay, that makes sense.
No, I was observing the following: mu is countably additive, and the set of theories is countable. Hence the measure of the total space is the sum of the measures of the theories, so the measures of the theories must sum to 1. Now it’s clear that at every step i of the process, the sum of the measures of the (incomplete) theories so obtained is 1. But it’s not immediately clear to me that this holds in the limit.
However, I just realized my mistake, which is that the set of theories isn’t always countable (there are countably many sentences, but a theory is a subset of the sentences; for instance, consider the language with countably many unary relations and a constant symbol). In particular, I believe it’s countable if and only if the sum is preserved in the limit, so we’re fine.
For a countable language L and theory T (say, with no finite models), I believe the standard interpretation of “space of all models” is “space of all models with the natural numbers as the underlying set”. The latter is a set with cardinality continuum (it clearly can’t be larger, but it also can’t be smaller, as any non-identity permutation of the naturals gives a non-identity isomorphism between different models).
Moreover this space of models has a natural topology, with basic open sets {M: M models phi} for L-sentences phi, so it makes sense to talk about (Borel) probability measures on this space, and the measures of such sets. (I believe this topology is Polish, actually making the space Borel isomorphic to the real numbers.)
Note that by Lowenheim-Skolem, any model of T admits a countable elementary substructure, so to the extent that we only care about models up to some reasonable equivalence, countable models (hence ones isomorphic to models on the naturals) are enough to capture the relevant behavior. (In particular, as pengvado points out, the Christiano et al paper only really cares about the complete theories realized by models, so models on the naturals suffice.)
I meant that (conjecturally) for every measure, there exists a cardinal kappa such that mu({M: |M| > kappa}) = 0. Anyway, I guess as you’ve demonstrated the set/class thing isn’t a big problem, but it is something to watch out for.
Okay, that makes sense.
No, I was observing the following: mu is countably additive, and the set of theories is countable. Hence the measure of the total space is the sum of the measures of the theories, so the measures of the theories must sum to 1. Now it’s clear that at every step i of the process, the sum of the measures of the (incomplete) theories so obtained is 1. But it’s not immediately clear to me that this holds in the limit.
However, I just realized my mistake, which is that the set of theories isn’t always countable (there are countably many sentences, but a theory is a subset of the sentences; for instance, consider the language with countably many unary relations and a constant symbol). In particular, I believe it’s countable if and only if the sum is preserved in the limit, so we’re fine.
For a countable language L and theory T (say, with no finite models), I believe the standard interpretation of “space of all models” is “space of all models with the natural numbers as the underlying set”. The latter is a set with cardinality continuum (it clearly can’t be larger, but it also can’t be smaller, as any non-identity permutation of the naturals gives a non-identity isomorphism between different models).
Moreover this space of models has a natural topology, with basic open sets {M: M models phi} for L-sentences phi, so it makes sense to talk about (Borel) probability measures on this space, and the measures of such sets. (I believe this topology is Polish, actually making the space Borel isomorphic to the real numbers.)
Note that by Lowenheim-Skolem, any model of T admits a countable elementary substructure, so to the extent that we only care about models up to some reasonable equivalence, countable models (hence ones isomorphic to models on the naturals) are enough to capture the relevant behavior. (In particular, as pengvado points out, the Christiano et al paper only really cares about the complete theories realized by models, so models on the naturals suffice.)