Having divided the universe of sets into disjoint groups, it would be con- venient to attach a “number” to each collection which could be used the way natural numbers are used to refer to the sizes of finite sets. Given a set X, there exists something called the cardinal number of X, denoted cardX, which behaves very much in this fashion. For instance, two sets X and Y satisfy cardX = cardY if and only if X ∼ Y. (Rigorously defining cardX requires some significant set theory. One way this is done is to define cardX to be a very particular set that can always be uniquely found in the same equivalence class as X.)
Looking back at Cantor’s Theorem, we get the strong sense that there is an order on the sizes of infinite sets that should be reflected in our new car- dinal number system. Specifically, if it is possible to map a set X into Y in a 1–1 fashion, then we want card X ≤ card Y . Writing the strict inequality card X < card Y should indicate that it is possible to map X into Y but that it is impossible to show X ∼ Y . Restated in this notation, Cantor’s Theorem states that for every set A, card A < card P (A).
There are some significant details to work out. A kind of metaphysical problem arises when we realize that an implication of Cantor’s Theorem is that there can be no “largest” set. A declaration such as, “Let U be the set of all possible things,” is paradoxical because we immediately get that cardU < cardP(U) and thus the set U does not contain everything it was advertised to hold. Is- sues such as this one are ultimately resolved by imposing some restrictions on what can qualify as a set. As set theory was formalized, the axioms had to be crafted so that objects such as U are simply not allowed. A more down-to- earth problem in need of attention is demonstrating that our definition of “≤” between cardinal numbers really is an ordering. This involves showing that cardinal numbers possess a property analogous to real numbers, which states that if card X ≤ card Y and card Y ≤ card X, then card X = card Y . In the end, this boils down to proving that if there exists f : X → Y that is 1–1, and if there existsg:Y →X that is 1–1, then it is possible to find a function h:X→Y that is both 1–1 and onto.
A proof of this fact eluded Cantor but was eventually supplied independently by Ernst Schr ̈oder (in 1896) and Felix Bernstein (in 1898).
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