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| | Infinitary Logic |
 | | Given a pair κ, λ of infinite cardinals such that λ ≤ κ, we define a class of infinitary languages in each of which we may form conjunctions and disjunctions of sets of formulas of cardinality < κ, and quantifications over sequences of variables of length < λ. |
| | It was quickly realized that a measurable cardinal must be inaccessible, but the falsity of the converse was not established until the 1960s when Tarski showed that measurable cardinals are weakly compact and his student Hanf showed that the first, second, etc. inaccessibles are not weakly compact (cf. |
| | Although the conclusion that measurable cardinals must be monstrously large is now normally proved without making the detour through weak compactness and infinitary languages, the fact remains that these ideas were used to establish the result in the first instance. |
| plato.stanford.edu /entries/logic-infinitary (6663 words) |
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