TITLE: Boolean Valued Models of PA
QUESTION [14 upvotes]: O.K, a massively naive question. I've never really studied any non-standard models of PA before. I was just wondering if there's ever been any attempt to use the kind of Boolean valued model theory familiar from set theory in the context of arithmetic. Obviously, the actual structure of the model would have to be very different, but I mean specifically the idea of using elements of a complete Boolean algebra as truth-values for sentences of PA. If this hasn't been done, has anyone got any ideas about if/how it could be done?

REPLY [9 votes]: This belated answer is prompted by:
(1) Joel Hamkins' answer, in which he brings attention to Boolean ultrapowers of models of arithmetic.
(2)  Emil Jeřábek's comment "Using sophisticated tools to construct elementary extensions is simply a waste of effort" (in the exchange with Hamkins following Jeřábek's answer).
First a definition: full arithmetic is the first order theory of the full expansion of the standard model of arithmetic, i.e, it is the first order theory of the model $ (\Bbb{N}, X)_{X \subseteq{\omega}}$, where $\Bbb{N}$ is the standard model of arithmetic $(\omega, +, \times)$.  Note that full arithmetic has an uncountable vocabulary.
Next, two results of Andreas Blass from the mid-1970s:
Theorem 1. Every model $\cal{A}=$ $(A, \cdot \cdot \cdot) $ of full arithmetic has a conservative elementary end extension $\cal{B}=$ $(B, \cdot \cdot \cdot) $, i.e., if X is a subset of B that is parametrically definable in $\cal{B}$, then $X \cap A$ is parametrically definable in $A$.
Theorem 2. Assuming CH (the continuum hypothesis) there is a model of full arithmetic that possesses a nonconserative elementray end extension.
And, finally, the point of this posting: back in 1990, I used the method of Boolean ultrapowers to improve theorem 2 above by eliminating the CH assumption.
My result appears as Theorem 3.4 of the paper below, which includes references for the aforementioned results of Blass, as well as other applications of the method of Boolean ultrapowers.
A. Enayat, Minimal elementary extensions of models of set theory and arithmetic. 
Arch. Math. Logic 30 (1990), no. 3, 181–192.