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Tangential to your point about the poor phrasing is that having "infinite ... numbers between just 0 and 1" is not evidence of a higher infinity. Specifically,
by andrewla 2y ago
Tangential to your point about the poor phrasing is that having "infinite ... numbers between just 0 and 1" is not evidence of a higher infinity.
Specifically, there are infinite rational numbers between 0 and 1, but rationals have the same cardinality as the natural numbers.
- rvbissell 2y agoThat's an interesting statement about rationals. My intuition would be to consider them a subset of the reals, and for natural numbers to be a subset of the rationals. How is my intuition failing me, in this case? EDIT: found the answer: with Cantor's diagonalization, you can count all the rationals -- effectively mapping each one to a natural number. Since this mapping is demonstrably possible, they have the same cardinality.