> But Cantor discovered that natural numbers can’t be put into one-to-one correspondence with the continuum of real numbers. For instance, try to pair 1 with 1.00000… and 2 with 1.00001…, and you’ll have skipped over infinitely many real numbers (like 1.000000001…). You can’t possibly count them all; their cardinality is greater than that of the natural numbers.
I take issue with this paragraph. It makes it sound like the infinitude of reals between 1 and 1.00001 is proof there are more reals than naturals. One might think this same fact proves that there are more rationals than naturals...
I take issue with this paragraph. It makes it sound like the infinitude of reals between 1 and 1.00001 is proof there are more reals than naturals. One might think this same fact proves that there are more rationals than naturals...