The Cantor conception of "infinity" is bad and should be retired. It's a pure naval-gazing exercise that has set back practical mathematics significantly. The notion of the "continuum" is fundamentally flawed. Intuitionalism and constructivism are a much better framework for reasoning about mathematics.
The response to the idea that the natural numbers and the even numbers and the rational numbers and the computable numbers have the same "cardinality" should be "who cares". All it means is that Cantorian cardinality is a bad standard against which to judge the "size" of an infinite set.
It ends in the same trap as the Axiom of Choice, where you can produce lots of obviously incorrect results and remark about how amazing they are; similarly you can make nonsensical statements about things being true "almost everywhere" that are clearly actually true nowhere.
The response to the idea that the natural numbers and the even numbers and the rational numbers and the computable numbers have the same "cardinality" should be "who cares". All it means is that Cantorian cardinality is a bad standard against which to judge the "size" of an infinite set.
It ends in the same trap as the Axiom of Choice, where you can produce lots of obviously incorrect results and remark about how amazing they are; similarly you can make nonsensical statements about things being true "almost everywhere" that are clearly actually true nowhere.