  We construct the following definition. Let κ be an infinite cardinal. A language L is said to be κ-compact (resp. weakly κ-compact) if whenever Δ is a set of L-sentences (resp. a set of L-sentences of cardinality ≤ κ) and each subset of Δ of cardinality &lt; κ has a model, so does Δ. Notice that the usual compactness theorem for  L  is precisely the assertion that  L is ω-compact. One reason for  according significance to the κ-compactness property is the  following. Call L κ-complete  (resp. weakly κ-complete) if there is a  deductive system P for L with deductions of length &lt; κ such that, if Δ is a P-consistent[4]  set of L-sentences (resp. such that |Δ| ≤ κ), then Δ has a model. Observe that such a P will be adequate for deductions from arbitrary sets of premises (of cardinality ≤ κ) in the sense of §2. It is easily seen that if L is κ-complete or weakly κ-complete, then L is κ-compact or weakly κ-compact. Thus, if we can show that a given language is not (weakly) κ-compact, then there can be no deductive system for it with deductions of length &lt; κ adequate for deductions from arbitrary sets of premises (of cardinality ≤ κ).