I think the mystery is not that some infinities can be larger than others, but that there are infinite sets of equal ‘size’ that conflict with intuition.
Some examples that ‘prove’ some infinities are larger than others to laymen:
- there are twice as many integers as odd integers
- there are more points on a plane then on a line
- there are more points on a line than on a circle
- there are more points on a plane then on a semiplane
- there are more rationals than integers
- there are more reals than rationals
It’s only in the intermediate state of a mathematician’s education, where they have just accepted that, for infinite sets, ‘more’ isn’t the best way to determine size equivalence that it becomes a surprise that for the last one, “the size of the set of reals is larger than that of the rationals” is true, and can be proven to be.
Some examples that ‘prove’ some infinities are larger than others to laymen:
- there are twice as many integers as odd integers
- there are more points on a plane then on a line
- there are more points on a line than on a circle
- there are more points on a plane then on a semiplane
- there are more rationals than integers
- there are more reals than rationals
It’s only in the intermediate state of a mathematician’s education, where they have just accepted that, for infinite sets, ‘more’ isn’t the best way to determine size equivalence that it becomes a surprise that for the last one, “the size of the set of reals is larger than that of the rationals” is true, and can be proven to be.