Cantor thought that he could measure something that has no size. He looked at the infinity of the natural numbers and noted it follows a particular pattern. He called that pattern aleph-zero, claimed it was a number and that it was possible to do arithmetic with it. He then proceeded to examine the infinity of reals, noted that they cannot be put into one-to-one correspondence with the naturals hence had a different fundamental pattern.

What Cantor was doing was identifying patterns or the organisation of the abstract structures formed in our minds by different types of infinity. Patterns are not sizes/cardinalities/numbers. It is, in general, not possible to do arithmetic with patterns. What is a snowflake plus one? What is a hexagon times a snowflake? - nonsensical questions.

Infinity is just an abstract concept that exists in our minds. It takes two different organisations/patterns (that of the natural and the reals). It is not measurable because it conceptually goes on forever. It has no size/cardinality because it is unmeasurable. Cantor identified that there are two different such abstract structures with different patterns/layouts. That is the sum of his legacy and no more.