Are there any natural measures that can be put on the space of models?

Question

I've already asked this question is Philosophy.SE and is a 'soft' question.

An undecidable proposition, as in Gödel's Incompleteness theroem, is one whose truth value cannot be determined, because it evaluates as true in some models, and false in others.

One could argue that this is an opening for a more complex notion of truth.

Is there some natural way of establishing a magnitude for this, so perhaps expanding the binary notion of truth (false/true) into a more sophisticated one. Perhaps if a natural probability measure is available on the space of models?

Answer 1

I don't think Gödel's incompleteness theorem (mentioned in the question) has much to do with this. You don't need to get anywhere near Gödel's incompleteness theorem to know that some statements are true in some models and false in others. For example, the existence of multiplicative inverses of elements other than $0$ is true in $\mathbb Z\bmod n$ if $n$ is prime, and false if $n$ is composite. That's taught in undergraduate courses to students who may never have heard of Gödel.

One can put probability measures on spaces of models, and then ask about the probability that a particular statement is true. But whether any is "natural" is clearly a much harder question.

Here's one small result I heard asserted once: Let's say you have a first-order language with finitely many relation symbols. If one regards all isomorphism classes of models of finite size $n$ as equally probable, then each statement in this language has some probability of being true in models of size $n$. And then for each statement one can take a limit as $n\to\infty$. A mathematician I spoke to claimed to have published this result: in every instance the limit is either $0$ or $1$. I don't remember his name.

I've never heard any such results for infinite models.

Answer 2

"An undecidable proposition, as in Gödel's Incompleteness theorem, is one whose truth value cannot be determined."

That's not quite right. If $G$ is a Gödel sentence for theory $T$, then to be sure $T$ can't settle whether $G$ or not-$G$. But some other theory might well be able to do just that. For example, some quite modest theories -- ones that almost any mathematician believes -- determine that the canonical Gödel sentence for first-order Peano arithmetic is true.