Packet 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 (15[1])theorem. (15[1])Unions with this property are implied by the prefix sigma, (15[1])as in a (15[1])sigma-finite measure. Non-negativity, (-5[1])assigning zero to the null set, and a type of (*) additivity named for this property (10[1])are the axioms of a measure. (10[1])A famous proof by contradiction from 1891 (10[1])assumes that a set [emphasize] has this property, (10[1])then forms a new element that differs (10[1])from all (10[1])others in one coordinate. (-5[1])This property is [emphasize] not held by the set of infinite binary sequences, (10[1])by Cantor’s diagonal (10[1])argument. (10[1])For 10 points, name this property of a set that has the same cardinality as the natural numbers. ■END■ (10[2])

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