- #1
dodo
- 697
- 2
Hello, I have a curiosity.
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?
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?