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I am currently working through the book "Model Theory" by Chang and Keisler and now I am stuck at exercise 2.1.2, which states:

"Prove the representation theorem for Boolean algebras by the method of diagrams."

For the authors the mentioned theorem is: "Every Boolean algebra is isomorphic to a field of sets, i.e. to a subalgebra of the power-set algebra of some set."

They also give hints:

  1. Every atomic Boolean algebra is isomorphic to a field of sets.
  2. Every finite subset of a Boolean algebra generates a finite, therefore atomic, Boolean algebra.
  3. If a Boolean algebra is isomorphically embedded in a field of sets, then it is isomorphic to a field of sets.

I managed to proof all three hints and with the following assumption I easily can proof the theorem by the method of diagrams:

Assumption: There exists a first order theory $T$ of the language $\mathcal{L}$ of Boolean algebras such that for every $\mathcal{L}$-structure $\mathfrak{F}$ the following holds: $\mathfrak{F}$ is (isomorphic to) a field of sets if and only if $\mathfrak{F} \models T$.

Now my question is wheither this statement holds. If yes, how can one see it or even construct such a theory $T$? If no (what I am expecting), how can we proof the theorem by the method of diagrams using the hints?

Thank you in advance for any help!

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The trick you're missing is to find an embedding into an atomic Boolean algebra, not a field of sets. Atomic Boolean algebras definitely are the models of a first-order theory (i.e., your assumption holds if yoou replace "field of sets" with "atomic Boolean algebra"), and so you can embed any Boolean algebra into an atomic Boolean algebra. But since any atomic Boolean algebra is a field of sets, this also proves you can embed any Boolean algebra into a field of sets.

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  • $\begingroup$ Thank you so much! $\endgroup$
    – Daniel W.
    Commented Dec 18, 2019 at 16:31

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