User blog comment:Edwin Shade/A Question Concerning Cardinal Infinities/@comment-24920136-20171025011457/@comment-5529393-20171025071752

Cardinality is a technical term.

An injection from A to B is a function from A to B such that no two elements of A map to the same element in B. So every element of A maps to a different element of B, but some elements in B may not get mapped to.

A bijection from A to B is a functino from A to B such that no two elements of A map to the same element in B, and every element in B is mapped to by some element in B. Basically, we have a one-to-one correspondence between the elements of A and the elements of B.

We say that A and B are equipollent, or have the same cardinality if there is a bijection from A to B. (or equivalently, a bijection from B to A)

We say that the cardinality of A is less than the cardinality of B if there is an injection from A to B, but no injection from B to A.

Now, to me, this is a good characterization of the notion of "size". If, say, we have 3 apples and 3 oranges, I can pair off the apples and the oranges perfectly. But, if I have 3 apples and 5 oranges, then no matter how I pair off apples and oranges, I will run out of apples before I run out of oranges.

This is quite similar to what happens for infinite sets and cardinality. For example, I can define a bijection from the natural numbers to the rational numbers, such that each natural number corresponds to a unique rational number and vice versa. So it is natural to think of them as being of the same "size".

On the other hand, if I take the natural numbers and the real numbers, I can define injections from the natural numbers into the real numbers. But, by Cantor's diagonal argument, there will always be reals left over. So I use up all the natural numbers, but I always have more reals untouched - it seems natural to say then that there are "more" reals than natural numbers.

So, to me, cardinality is a good definition of "size" for sets, and certainly the best one we have. Now, you are free to disagree with this. To you, perhaps it is obvious that all infinite sets are the same "size". This is your own intuitive interpretation of a nonmathematical term, so it's not wrong. But, mathematicians cannot be wrong for defining "cardinality" however they choose - it doesn't have to agree with anyone's particular notion of anything. Certainly, cardinality agrees with our notion of size for finite sets, and naturally extends it to include infinite sets. And, I'm sure you will agree, cardinality is a more useful property of infinite sets than a definition of "size" in which all infinite sets are equal. So, there's no need for conflict here.