Weak compactness and small infinitary theories

If kappa is weakly compact, apply F2 to any theory in the assertion. F1 gives a cofinal branch in its truth tree, and F2 produces a model. Thus the L_(kappa,kappa) property holds. Every L_(kappa,omega) theory is a special case with finite blocks, so the latter compactness property follows.

Conversely assume the L_(kappa,omega) property and fix a kappa-tree S. It has kappa nodes: its kappa nonempty levels give the lower bound, and choosing injections of levels into kappa gives the upper bound by the cardinal-square estimate used in F2. Introduce a unary relation P_t for each node t and one constant d; write p_t for the sentence P_t(d). Take a theory consisting of the disjunction ${\bigvee }_{t\in S_{\alpha }}{p}_t$ for each alpha<kappa and the sentences $\mathrm{\neg }(p_s\wedge p_t)$ for each incomparable pair s,t. Every level disjunction has fewer than kappa terms, so this is already an L_(kappa,omega) theory (indeed it uses no quantifiers). Its language and theory have size at most kappa, using the same square bound.

Any subtheory of size below kappa has level-disjunction requirements at a bounded collection of levels, by regularity. Choose a node u above those levels. In a one-element structure interpret p_t as true exactly for t<=u. This meets every required level and violates no incomparable-pair prohibition, including prohibitions mentioning nodes at arbitrarily high levels. Hence every small subtheory is satisfiable. Compactness gives a full model. Its true p_t include a node at each level and never include incomparable nodes; they therefore form a cofinal branch of S. As S was arbitrary, F1 gives weak compactness. Together with step 1.1 this proves both equivalences.