Does every infinite set have a denumerable subset?
This question is answered with a mere "Yes" in the textbook (Theory of
Sets by Joseph Breuer), but I'm not sure where the confidence comes from.
For instance, if we have a non-denumerable set, then it seems to me that
no matter what pattern we choose, we'll end up with a subset with the same
cardinal number. Or won't we?
I guess what makes matters worse is that the question right after proving
the theorem that "If a denumrable set is subtracted from a non-denumrable
set, the resulting set is no-denumerable." I understood the proof, and can
even reproduce it perfectly, but I have no idea I understand what it
means. An example would make things clearer, I guess.
I'm thoroughly confused; please help!
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