Round 1: Tossup 20

If a partially ordered set meets a condition named for this property, any collection of dense subsets of small size gives rise to a generic filter, by Martin’s axiom. A consistent first-order theory has a model with this property, by the Löwenheim-Skolem theorem. Unions with this property are implied by the prefix sigma, as in a (15[1])sigma-finite measure. Non-negativity, assigning zero to the null set, and a type of (*) additivity named for this property are the axioms of a measure. (10[1])A famous proof by contradiction from 1891 assumes that a set [emphasize] has this property, (10[1])then forms a new element that differs from all others in one coordinate. This property is [emphasize] not held by the set of infinite binary sequences, by Cantor’s diagonal argument. For 10 points, name this property of a set that has the same cardinality as the natural numbers. ■END■

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