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?
 A: I'm not exactly sure what the supposed problem is, but I can walk through some basic results to hopefully shed light on what's going on.
Two basic notions in set theory are sets and functions. I'll assume you agree that the notion of a set and the notion of a function make sense.
Consider a function $f : A \to B$.
$f$ is said to be injective iff for all $x, w \in A$, if $f(x) = f(w)$ then $x = w$.
$f$ is said to be surjective iff for all $b \in B$, there exists $a \in A$ such that $f(a) = b$.
$f$ is said to be bijective iff $f$ is both surjective and injective.
We say that $A \preccurlyeq B$ if and only if there exists an injective function $f : A \to B$. We say $A \approx B$ if and only if there exists a bijective function $f : A \to B$.
Cantor's insight was that it is possible to assign each set $A$ something called a "cardinality". The cardinality of $A$ is denoted $|A|$. The assignment is done in such a way that for all sets $A$, $B$, we have $|A| = |B|$ if and only if $A \approx B$. The technical details of this are a bit complicated, but in the case that $A$ has exactly $n$ elements, $|A| = n$.
It turns out that once we define cardinality, we can also define the comparison operator $\leq$ on cardinalities. This operator is defined so that $|A| \leq |B|$ if and only if $A \preccurlyeq B$.
In practice, when one is first learning about cardinality, you should always take $|A| = |B|$ as a shorthand for $A \approx B$, and you should always take $|A| \leq |B|$ as a shorthand for $A \preccurlyeq B$.
We can prove explicitly that $\mathbb{Z}$ and $E$ have the same cardinality by considering the function $f : \mathbb{Z} \to E$ defined by $f(n) = 2n$.
To show that $f$ is a well-defined function, note that $E$ is defined to be $\{2n \mid n \in \mathbb{Z}\}$. Therefore, $f(n) \in E$ for all $n \in \mathbb{Z}$.
To show that $f$ is injective, consider $a, b \in \mathbb{Z}$ and suppose that $f(a) = f(b)$. Then $2a = 2b$. Dividing both sides by 2 gives $a = b$.
To show that $f$ is surjective, consider some $e \in E$. Then by the definition of $E$, there exists $n \in \mathbb{Z}$ such that $2n = e$. Then for this $n$, we have $f(n) = e$.
Therefore, $f : \mathbb{Z} \to E$ is a bijection. Thus, we have $\mathbb{Z} \approx E$. Using the notation of cardinalities, we have $|\mathbb{Z}| = |E|$.
There's nothing circular about any of it.
A: You are approaching cardinality and infinity in the wrong way. For two finite sets $A$ and $B$ the sets are of the same size if and only if there exists a bijection between the two sets. This is somewhat obvious, and the proof is done in the first few weeks of any discrete math or set theory course.
Of course we cannot think about infinite sets in the exact same way, but we use the same idea of a bijection to identify "families" or "classes" of infinite sets, and oftentimes we say that these equivalent classes of infinities have the same size; although that is not entirely accurate it gets the point across that these infinities exist within the same group, and share attributes.
As someone commented there is a clear bijection between integers and even numbers, you can try find a bijection between integers and natural numbers, or natural numbers and fractions on your own time to indeed convince yourself that these infinities are in the same "family" as we defined above. We call this "family" of infinities Countably Infinite.
You may be wondering what the point of this is...if we are just mapping primes, to natural numbers, to fractions, to even numbers, etc. that doesn't give us any extra information or really say anything about the sets, that is until you realise there are infinite sets that cannot be bijected to the natural numbers, a classic example is the Real Numbers, this is proven in Cantor's Argument of Diagonalization (One of the most elegant mathematical arguments ever in my opinion.)
A: 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.
