Question

Given an 2-valued measure $\mu$ on a set $I$, and structures $M_i \ (i \in I)$, one can construct the ultrapower $\prod_{i\in I}M_i / U$ (where $U$ is the ultrafilter associated with $\mu$.) One can then prove nice theorems about this structure - e,g, Łoś's theorem.

But suppose now that rather than beginning with a 2-valued measure, we begin with a real valued measure on the set $I$, and structures $M_i \ (i \in I)$. Is there any sort of analagous structure that we can build and prove useful results about?

Answer 1

There is a generalization of ultrapowers that actually gives something new, and this the socalled Boolean ultrapowers of [Mansfield, Richard, The theory of Boolean ultrapowers. Ann. Math. Logic 2 (1970/71), no. 3, 297–323] where you use ultrafilters in a Boolean algebra rather than on a set. That Boolean ultrapowers can be different from ordinary ultrapowers was shown in [Koppelberg, Bernd; Koppelberg, Sabine A Boolean ultrapower which is not an ultrapower. J. Symbolic Logic 41 (1976), no. 1, 245–249].

With real valued measures there are several problems that you can run into. First of all, if you require $\sigma$-additivity, you can usually not measure all subsets of the set $I$. So if for certain sentences $\varphi$ you want to assign a measure, namely the measure of the set of indices $i\in I$ such that $\varphi$ holds in $M_i$, you might not be able to do that in all cases. You would have to make sure that whether or not $\varphi$ holds in $M_i$ depends on $i$ in a measurable way. While this might look like not such a big deal, you will also need such a measurability condition for formulas with parameters. And then you have to think how you are allowed to choose parameters in the individual models. That should also happen in a way that is measurable in some sense.

Another problem is that given a (measurable) set $A\subseteq I$, it can happen that neither $A$ nor its complement are of measure $1$. This is going to cause trouble when you want to prove an analogue of Łoś's theorem. The structural induction that is necessary for the theorem just breaks down in various places.

You might be able to get a reasonable structure in terms of multivalued or fuzzy logic, but I am no expert on these things and it doesn't seem to be the case that you get something interesting in the sense of classical logic.

Answer 2

The reduced power and reduced product constructions are generalizations of the ultrapower construction that use filters instead of ultrafilters, and you can construct a filter from a measure(and hence a reduced power) simply by taking the collection of all sets whose complement is measure 0. See the book "A Course in Universal Algebra" at http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf for the definition of a reduced power.

There is a weak version of Łoś's theorem that holds for reduced powers although it is a bit messier for reduced powers which we shall state here. This result basically gives a method of determining the truth value of a sentence in a reduced power. In the book "Model Theory" by Chang and Keisler section 6.3, the author describes an effective procedure that does the following. Given any first order sentence $\phi(x_{1},...,x_{n})$ in some first order language $\mathcal{L}$, the algorithm finds a sequence $(\sigma,\theta_{1},...,\theta_{m})$ of formulas such that $\sigma(z_{1},...,z_{m})$ is a formula in the language of Boolean algebras, and each formula $\theta_{i}$ has at most the variables $x_{1},...,x_{n}$ free and where we have the following:

Assume $I$ is an index set, $Z$ is a filter on the set $I$, and $\mathcal{A}_{i}$ is an $\mathcal{L}$-structure for $i\in I$. Assume also that $f_{1},...,f_{n}\in\prod_{i\in I}A_{i}$, and given $1\leq j\leq m$, let $R_{j}=\{i\in I|\mathcal{A}_{i}\models\theta_{j}(f_{1}(i),...,f_{n}(i))\}$. Then we have \[\prod_{i\in I}\mathcal{A}_{i}/Z\models\phi(f_{1}/Z,...,f_{n}/Z)\] if and only if \[P(I)/Z\models\sigma(R_{1}/Z,...,R_{m}/Z).\]

I personally found the above result to be very useful since I was able to determine the sentence algebras and the elementary classes of a certain variety using a version of the above result. Of course, the above result is not as powerful as Łoś's theorem since it does not automatically give you a non-standard model of a structure like the real numbers.