Cantor's paradox is a counter-intuitive result (and thus a "benign" paradox) according to which the set of all sets does not exist. Contrary to modern axiomatic formulations of set theory, naive set theory views sets as collections of objects and works under the unspoken assumption that every natural language description of a set corresponds to a set. Even sets such as "the set of humans taller than ten feet" or "the set of all talking giraffes" exist; they are the empty set, which has no members (and should not be confused with null sets, which may not necessarily be empty, but that's a story for another time.)

The assumption that any set which we can describe must exist leads to many pitfalls. "The set of all sets which do not contain themselves" sounds like a perfectly viable set, but a little thought reveals that this set can neither belong nor not belong in itself. If it does belong to itself, it can't contain itself by its definition, and if it doesn't contain itself, it must belong in itself, again by definition. Clearly, something is wrong; sets are meant to divide the world into things that belong to them and things that don't. Something is indeed wrong, and it's the assumption that any set we can put into words is well-defined. The set of all sets which do not contain themselves is simply a spurious concept.

Back to the core of paradox. The set of all sets is also ill-defined, but for a slightly different reason. To understand what that is, we must take a look at a way to compare the "size" of different sets, since we expect that the set of all sets will also be the "largest" set that can possibly exist. Comparing finite sets is easy enough (the set of all children is clearly smaller than the set of all humans), but what about infinite sets? As one of my teachers used to say, if you didn't know how to count, you could tell that your hands have the same number of fingers simply by matching each finger on your left hand with exactly one finger on your right hand. Simply put, if we can match each and every element of a set with exactly one element of another set, then these two sets have the same number of elements, or cardinality for short.

You may have heard that there are as many even numbers as there are natural numbers, since we can match each number with an even one simply by doubling it, leaving no numbers out in the process. So, you may be tempted to think that infinity (the cardinality of the natural numbers) is the largest cardinality there is, and that would be the end of the story. It is, however, an incredibly surprising fact that there are infinite sets for which we cannot perform this exhaustive one-to-one matching between elements (technically called a bijection), indicating that, contrary to all expectations, one infinite set is larger than the other. Cantor's diagonal argument shows that the real numbers and the natural numbers are such an example; no matter how we match them, there will always be some real numbers left out, meaning that they more populous than the natural numbers.

So, having discovered this hierarchy of cardinal numbers, we expect to find our Holy Grail, the Set of All Sets, resting comfortably at the top. This, however, is where the crux of the "paradox" lies. There is no top; every set, no matter how large, is surpassed in cardinality by some other set. Thus, talking about the set of all sets is like talking about the largest number; both are equally meaningless. To outline a proof of why this is, we will need to introduce one more concept, that of a power set. The power set of a set is the set of all its subsets. An example is helpful: a three-member family {Dad, Mom, Son} has 8 subsets: {Dad, Mom, Son}, {Dad, Mom}, {Mom, Son}, {Dad, Son}, {Dad}, {Mom}, {Son}, {}, which make up the family's power set. It's obvious that this set is larger than the original, but does this hold for infinite sets?

Consider a universe populated by a number of citizens (note that we make no assumptions about the cardinality of the universe). The citizens belong to various clubs; there's a club for women, men, teenagers, people named John... in fact, every possible subset of citizens corresponds to some club. Do the universe and the set of clubs have the same cardinality? If they do, we expect to be able to make each and every citizen to act as representative for one and only one club so that no club is left without a representative. If the representative belongs the club they represent, we will call them (for administrative purposes) internal, and if they don't, we will call them external (some clubs' members are too shy to represent themselves, and of course the empty club will need to have an external representative).

Consider now, however, the club of all external representatives. Is the representative of this club internal or external? By now, you should be getting an eerie sense of deja vu; the reasoning is identical to case of the set of all sets which don't contain themselves. If the representative is internal to the club, he must be external by the very charter of the club that only admits external representatives, and if he is external, he must be internal to the club, as the club admits all external representatives. This club cannot exist.

Where did we go wrong? If the set of clubs contain all possible clubs, how can it not contain this one? The answer is that we defined a club based on the assumption that we can assign a unique representative to each club. By contradiction, we cannot. This means that the set of clubs (the power set in this extended analogy) has a different cardinality than the universe (the set). Proving that the power set has a strictly greater cardinality is not much more involved, but it is slightly more technical. Hopefully, it is intuitively clear: we can match each member x to their one-member club {x} and yet there are still clubs left without a representative.

So there you have it. The set of all sets does not exist as a direct consequence of Cantor's theorem, which is what we just sketched out: informally put, for every set, there is always a larger set. Besides necessitating the axiomatic development of a more rigorous set theory, Cantor's theorem has some interesting philosophical implications, such as the non-existence of the set of all true statements. As Patrick Grim points out in The Incomplete Universe, the set of all truths T = {t1, t2, ...} would have a power set made up of subsets S and for each truth ti, there would be an associated truth t'ij = "ti belongs/does not belong to subset Sj". But, of course, since we now know that there are more subsets S in the power set than there are truths, we see that the set of "meta-truths" T' is larger than T , and so T cannot contain all truths. Along with Gödel's incompleteness theorem, this result casts serious doubt on the optimistic view that every true statement is also knowable and that David Hilbert so poetically once expressed as "Wir müssen wissen, wir werden wissen!" ("We must know, we will know").