- #1

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With a finite number of bits, say 5, you can represent 2^5 items. If B = {00001, 00010, 00100, 01000, 10000} is the set of these 5 bits, you can combine them to represent the power set of B, with cardinality 2^5.

With a countably infinite number of bits, you could represent a countable set like the naturals, using only one bit per natural, as in {...00001, ...00010, ...00100, ...01000, ...}; it would appear that by combining bits you could represent the power set of the previous, with cardinality aleph_1.

Thus it would look like, to represent the naturals with bit combinations, you'd need an infinite,

*yet less than countable*, number of bits, such that its power set is countable. Where is the way out of this paradox?