You can tell two piles are equal without counting either one
Lay out a heap of apples and a heap of oranges, then match them off one apple to one orange. If nothing is left over, the heaps are the same size, and you never needed a single number. That simple pairing trick is how mathematicians compare infinite collections, and it reveals that some infinities are bigger than others.
Cardinality is the size of a set, and pairing is its measuring stick. Two sets match in size when every member of one can be given a partner in the other with nobody left alone and nobody paired twice; mathematicians call such a pairing a bijection. For finite sets this just recovers ordinary counting. For infinite ones it produces surprises, because the old rule that a whole must be larger than any of its parts stops working.
The even numbers 0, 2, 4, 6 and onward look like half of all natural numbers, yet doubling pairs every natural number with exactly one even number, so the two sets have the same cardinality. The rational numbers turn out the same way. Any set that can be lined up against the natural numbers like this is called countable. The real numbers cannot: so-called diagonal arguments prove they are strictly larger, and Cantor's theorem extends the reasoning to an endless ladder of ever larger infinities, usually labelled with the Hebrew letter aleph.
Whether the real numbers occupy the very next rung above the natural numbers is the continuum hypothesis, and it can be neither proved nor disproved from the standard Zermelo–Fraenkel axioms. Hints of these ideas go back to the sixth century BC, but earlier thinkers who stumbled on them usually wrote them off as paradoxes.
Georg Cantor formally introduced the concept in the late nineteenth century under the German word Mächtigkeit, meaning magnitude or power, a term he said he took from Jakob Steiner's work on projective geometry. The English words cardinality and cardinal number took over around 1930. Cardinal itself comes from Latin cardo, a hinge, the same root behind cardinal sins and the cardinal points of the compass.
Source: Cardinality
