Strong compactness, fine measures and infinitary logic

Assume filter extension and fix lambda>=kappa. For a in P_kappa(lambda), let C_a be the cone of x containing a. The sets containing some C_a form a proper filter: every cone contains a itself, and C_a intersect C_b=C_(a union b). An intersection of eta<kappa such filter members contains the cone above the union of their witnessing a_xi. F3 selects those witnesses; regularity makes their union have size below kappa (bound all sizes below one cardinal below kappa and use the infinite-cardinal product bound). Thus the filter is kappa-complete. Extend it by F1. Since C_{alpha} belongs to it for each alpha<lambda, the extension is fine.

Assume the fine-measure assertion and fix a theory T as in the L_(kappa,kappa) assertion. Choose a cardinal lambda>=kappa and an injection e:T to lambda. Push a fine measure on P_kappa(lambda) forward by $x\mapsto e^{-1}[x]$ to get a proper kappa-complete ultrafilter W on P_kappa(T). Inverse images preserve all intersections and complements, proving these laws. For every t in T, the inverse image of the cone of small subtheories containing t contains the e(t)-cone, so W is fine.

Each small subtheory a has a set model in the common signature. For each a take the least rank rho_a of such a model code. This is uniquely defined by set satisfaction, and Replacement collects the ranks. Separation in V_(rho_a+1) gives the nonempty set of rank-rho_a model codes satisfying a; F3 chooses one for each a. This bounds the choices before AC and does not choose from classes of models. Take their set ultraproduct by W. For each t in T the coordinate truth set contains the cone of a containing t, hence is in W. F2 makes t true in the ultraproduct. Thus T has a model. The L_(kappa,omega) assertion follows by restriction of syntax.

Assume the L_(kappa,omega) assertion and let F be a proper kappa-complete filter on a set I. Introduce one constant d and one unary relation R_X for every subset X of I; abbreviate R_X(d) by r_X. Include r_I, not r_empty, the equivalences $r_{I\setminus X}\leftrightarrow \mathrm{\neg }{r}_X$, and $r_{{\bigcap }_{\xi <\eta }{X}_{\xi }}\leftrightarrow {\bigwedge }_{\xi <\eta }{r}_{X_{\xi }}$ for every eta<kappa and every such sequence of subsets. Also include r_X for each X in F. This is a set theory in L_(kappa,omega), using no quantifier blocks. For any subtheory of size below kappa, the fewer-than-kappa F-members explicitly required have a nonempty intersection by completeness and properness. Take i in it and interpret the predicates on a one-element carrier by r_X true exactly when i belongs to X. Every identity axiom is then true, even those mentioning other subsets, and all the chosen F-requirements hold. Empty collections use I, which is nonempty since F is proper.

Compactness yields a full model. Define U to be the subsets X for which r_X is true there. The top and bottom axioms give properness. The intersection identities give kappa-completeness; their binary instances also give upward closure, since X subset Y implies X intersect Y=X and r_X forces r_Y. Complement identities decide exactly one of X and its complement. Thus U is a kappa-complete ultrafilter, and the F-requirements give F subset U. This proves filter extension, closing the cycle of equivalences.