adding upon this comment to why the two cardinalities are not equal, on one hand we have the set of integers {..., -2, -1, 0, 1, 2, ...} and they can be put into a bijection with the set of natural numbers {1, 2, 3, 4, ...}, this is done by rearranging the set of integers like {0, -1, 1, -2, 2, -3, 3, ...}. so this is a countably infinite set (one that has a cardinality of ℵ_0)
As for the set of real numbers, we have the subset of irrational numbers which are uncountably infinite (see cantors diagonalization argument) thus making the whole set of real numbers, a set whose cardinality is ℵ_1.
The annotated turing book goes into this pretty well in the first couple pages.
As for the set of real numbers, we have the subset of irrational numbers which are uncountably infinite (see cantors diagonalization argument) thus making the whole set of real numbers, a set whose cardinality is ℵ_1.
The annotated turing book goes into this pretty well in the first couple pages.