About continuum hypothesis

Question

Sir why continuum hypothesis isn't falsifiable. It states that "There is no set whose cardinality is strictly between that of the integers and the real numbers"; so can't we make a set which includes all integers and one irrational number say sqrt(2), and name this set X. So this set (X) would have one element more than the set of integers (i.e sqrt(2)), but would have infinite elements less than the set of real number.

I know that I am wrong, but please lest me understand this.

Answer 1

The mistake in your argument is that for any given infinity the quantity infinity+(1 element)= the same "order" of infinity. So you can not increase the infinite cardinality of a set by adding just one element.