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 model with this property can nonetheless possess sets (emphasize) without this (15[1])property, by (15[1])Skolem’s (15[1]-5[1])paradox. (-5[1])Unions with this property are (15[1])implied by the prefix sigma, as in a (15[1])sigma-finite (15[1]-5[1])measure. Non-negativity, assigning (15[1])zero to the (-5[1])null set, and a type of (*) additivity named for this (10[2]-5[1])property are the (10[1])axioms of a measure. A famous proof by contradiction from 1891 assumes that a set [emphasize] has this property, then forms a (-5[1])new element that differs from all (10[1])others in one (10[2])coordinate. This property is [emphasize] not (-5[1])held by the set of infinite binary sequences, by Cantor’s diagonal (10[2])argument. For (10[1])10 (10[1])points, name this property of a set that has the same (10[1])cardinality as the natural numbers. (10[6]0[1])■END■

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