Pete's excellent notes have correctly explained that there is no set containing sets of unboundedly large size in the infinite cardinalities, because from any proposed such family, we can produce a set of strictly larger size than any in that family.
This observation by itself, however, doesn't actually prove that there are uncountably many infinities. For example, Pete's argument can be carried out in the classical Zermelo set theory (known as Z, or ZC, if you add the axiom of choice), but to prove that there are uncountably many infinities requires the axiom of Replacement. In particular, it is actually consistent with ZC that there are only countably many infinities, although this is not consistent with ZFC, and this fact was the historical reason for the switch from ZC to ZFC.
The way it happened was this. Zermelo had produced sets of size $\aleph_0$, $\aleph_1,\ldots,\aleph_n,\ldots$ for each natural number $n$, and wanted to say that therefore he had produced a set of size $\aleph_\omega=\text{sup}_n\aleph_n$. Fraenkel objected that none of the Zermelo axioms actually ensured that $\{\aleph_n\mid n\in\omega\}$ forms a set, and indeed, it is now known that in the least Zermelo universe, this class does not form a set, and there are in fact only countably many infinite cardinalities in that universe; they cannot be collected together there into a single set and thereby avoid contradicting Pete's observation. One can see something like this by considering the universe $V_{\omega+\omega}$, a rank initial segment of the von Neumann hierarchy, which satisfies all the Zermelo axioms but not ZFC, and in which no set has size $\beth_\omega$.
By adding the Replacement axiom, however, the Zermelo axioms are extended to the ZFC axioms, from which one can prove that $\{\aleph_n\mid n\in\omega\}$ does indeed form a set as we want, and everything works out great. In particular, in ZFC using the Replacement axiom in the form of transfinite recursion, there are huge uncountable sets of different infinite cardinalities.
The infinities $\aleph_\alpha$, for example, are defined by transfinite recursion:
- $\aleph_0$ is the first infinite cardinality, or $\omega$.
- $\aleph_{\alpha+1}$ is the next (well-ordered) cardinal after $\aleph_\alpha$. (This exists by Hartog's theorem.)
- $\aleph_\lambda$, for limit ordinals $\lambda$, is the supremum of the $\aleph_\beta$ for $\beta\lt\lambda$.
Now, for any ordinal $\beta$, the set $\{\aleph_\alpha\mid\alpha\lt\beta\}$ exists by the axiom of Replacement, and this is a set containing $\beta$ many infinite cardinals. In particular, for any cardinal $\beta$, including uncountable cardinals, there are at least $\beta$ many infinite cardinals, and indeed, strictly more.
The cardinal $\aleph_{\omega_1}$ is the smallest cardinal having uncountably many infinite cardinals below it.
As your quotes from Berlinski suggest, the heart of Godel's statements is arguably not in the unprovability itself, but in the fact that the system is able to 'talk about itself' - that is, that statements about provability can be cast as strictly arithmetic questions. It might help to familiarize yourself with another undecidability result which goes through roughly the same route: Matiyasevich's Theorem about the solvability of diophantine equations (i.e., the question 'does this polynomial in $x$, $y$, $z$, $w$, etc. ever evaluate to zero at some integer values of $x$, $y$, $\ldots$?'). The root of the proof lies in showing that diophantine equations are expressive enough to represent all recursively enumerable sets; what Godel does is essentially the same, showing that Peano Arithmetic (or any similar system) is expressive enough that questions about 'provability' can be cast as arithmetic questions.
As for why unprovable statements need admission, the point isn't simply that they're unprovable but that they're unprovable and true - that is, that the set of 'things we can prove' will always be a proper subset of 'things that are true'. Obviously this isn't a crippling discovery; mathematics didn't stop as of Godel! But it is arguably a profound one, and one that has substantial practical implications: for instance, as noted above there can't possibly be an algorithm for solving diophantine equations; there can't be algorithms for finding a 'minimal forbidden minor' categorization of many sets of graphs; there can't be algorithms for solving the Word Problem for groups; etc, etc.
Best Answer
That's correct, but it is not very interesting: we don't have "access" to most mathematical facts. It is not even clear what a proof of an arbitrary mathematical fact would mean, because to ask for a proof of something we have to write it down, and there are only countably many strings we can write.
Gödel's theorem is much more interesting: it says that there is a formula that is independent of PA. So you can write some statement in arithmetic language (and it actually can be as simple as "this Diophantine equation has a solution") and neither it nor its negation can be proved in PA.
Compare it with, say Euclidean geometry. There are uncountably many lines and points, so there are an uncountable number of "facts" like "this point is / is not on this line". But Euclidean geometry is complete. So Gödel's theorem shows a difference between arithmetic and geometry.
And also there is (assuming consistency of ZF) a model of ZF where there are, from an outside point of view, only countably many sets.