I know little about theory, so the following maybe a stupid question .
"one-to-one correspondence"(bijection) is a good method to judge whether two finite sets have the same size , while a lot of mathematical rules become invalid in the world of infinite , so
(1).how Cantor make it sure that bijection of two infinite sets can also ensure the two infinite sets have the same size(or the same number of elements) ?
(2). What "one-to-one correspondence"(bijection) is used for in set theory ?
P.S. The "one-to-one correspondence"(bijection) caused many unexpected surprising result with infinite sets , maybe wrong result from my perspective , like all positive integers and its subset all positive odd numbers have the same cardinality, the same size(or the same number of elements), however, lots of people think the number of positive integers should be twice of its subset all positive even numbers before accepted Cantor's conclusion about this. Counter-intuitive result caused when comparing the size of two infinite sets using bijection cannot assert comparing the size of two infinite sets using bijection is wrong , but I think if we find another standard to compare the size of two infinite sets and also make the result useful and accord with our intuition, that would be much better .
Best Answer
First you ask
The answer, which you seem to understand, is that he didn't "make it sure", he made that the definition of "have the same size".
Then you note, correctly, that this definition leads to many counterintuitive results, adding
Perhaps. But mathematicians haven't succeeded in finding another definition that's as generally useful. They've taken an alternative path, refining their intuition about "same size" so as to give results they can prove from Cantor's definition.
You might want to read about some of the controversy surrounding Cantor's work. Some mathematicians at the time tried to do what you suggest. David Hilbert championed Cantor:
(From https://en.wikipedia.org/wiki/Cantor%27s_paradise)