  (3.2) κ is weakly compact ⇒  L(κ,ω)  is weakly κ-compact ⇒ κ is weakly inaccessible. Accordingly, it is consistent (with the usual axioms of set theory) to assume that no language  L(κ,ω)  with κ ≥ ω1 is weakly κ-compact, or, a fortiori, weakly κ-complete.