Basic set theory question As I understand it, Cantor defined two sets as having the same cardinality iff their members can be paired 1-to-1.  He applied this to infinite sets, so ostensibly the integers (Z) and the even integers (E) have the same cardinality because we can pair each element of Z with exactly one element of E.
For infinite sets, this definition seems problematic no matter which direction we come at it from:  We don't know up front that two infinite sets have the same cardinality, so we cannot conclude that their elements can be exactly paired.  And we do not know up front that two infinite sets' elements can be paired up exactly (because we don't know with certainty what happens beyond the finite cases we can verify).  So we cannot conclude that their cardinalities are the same.  The definition above therefore seems useless, since we cannot start from either side of the "iff".
It might be argued that if we state it as follows:  "For each element e of E, pair it with element e/2 of Z," then we have expressed the general case symbolically, and it works.  But we can only verify that for finite values of E and Z.  We can't know what happens beyond finite elements of those sets.  So expressing it symbolically does not seem to help.
Why is Cantor's definition not circular and therefore useless for deciding the question of infinite set cardinalities?

There are many definitions where you kind of need to know the answer before you use the definition. I write "kind of" because you don't need to have a proof for it, but you need to have a guess. Then you write a proof to show that the guess is actually correct.
The definition of equal cardinality is such a definition. Another one that you might have seen is the definition of limit: $\lim_{x\to a} f(x) = L$ if for all $\epsilon>0$ there exists $\delta>0$ such that $|f(x)-L|<\epsilon$ whenever $|x-a|<\delta.$ Here you need to have a guess of the limit $L$ before you can show that $\lim_{x\to a} f(x) = L.$ But primarily the definition is used to show some laws/rules. These laws can then be used to determine a limit and no explicit use of the definition is needed. Similarly one can find laws for cardinalities and use them instead of the definition.
