- #1

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The cardinality of all subsets of the integers is [itex] \aleph_1 [/itex]. A subset of the integers can be identified with 1-1 mapping of the integers onto the set consisting of only two integers {0,1}. So the cardinality of M should be at least that large.

A 1-1 function is a set of ordered pairs of numbers that satisfy certain conditions. The set of all ordered pairs of integers has a cardinality [itex] \aleph_0 [/itex]. The set S of all possible subsets of that set has cardinality [itex] \aleph_1 [/itex]. The set M is proper subset of S so the cardinality M should be at most [itex] \aleph_1 [/itex].

Perhaps I need some famous "optional" assumption of mathematics, such as the continuum hypothesis, to justify some of those statements.