Georg Cantor was a brilliant
nineteenth century mathematician whose discoveries led to the foundation of mathematics
becoming axiomatic set theory.

^{1}Cantor’s paradox concerns the collection of all the sets. Now, a collection is just some things being referred to collectively, of course, and a set is basically a non-variable collection. (Sports clubs and political parties are variable collections, for example, while chess sets and sets of stamps are non-variable.) And numbers – non-negative whole numbers – are basically properties of sets. Cantor’s core result was an elegant proof that even infinite sets have more subsets than members (in the cardinal sense of ‘more’). It follows that if there was a set of all the other sets, then it would have more subsets than members, whence there would be more sets than there are sets – each subset being a set – which is impossible. And it follows that there is no set of all the other sets. That consequence is known as ‘Cantor’s paradox’; but, how paradoxical is it? Presumably {you} did not exist until*you*did, so why expect the collection of all the sets to be non-variable?
A more paradoxical consequence
is, I think, that there is no set of all the numbers – for a proof of that
consequence, see section 3 of my earlier Who's Afraid of Veridical Wool? – which is paradoxical because
we naturally think of numbers as timeless, whence their collection should be
non-variable. To see the problem more clearly, suppose that 0, 1, 2 and 3 exist,
but that as yet 4 does not; the problem is: How could 2 exist, but not two
twos? The existence of

*n*(where ‘*n*’ stands for any whole number) amounts to the existence of the possibility of*n*objects, e.g.*n*tables (were physics to allow so many), and possibilities are, intuitively, timeless: For anything that exists, it was always possible for it to exist.
Nevertheless, if there
are too many numbers for them all to exist as distinct numbers, perhaps they
are forever emerging from a more indistinct coexistence. Possibilities are not
necessarily timeless. You were always possible, for example, but that
possibility would – had you never existed – have been the possibility of

*someone just like you*. Looking back, there was always the possibility of*you yourself*, as well as that more general possibility; but had you not existed, there could have been no such distinction. Note that if the universe had bifurcated into two parallel universes, identical in all other respects, then the other person just like you would not have been you. And however many parallel universes there were, another would not appear to be logically impossible. So, it appears to be logically possible for apparently timeless possibilities – e.g. the possibility of you yourself – to emerge as distinct possibilities from more general possibilities.
Those taking numbers to
be timeless also have some explaining to do: They need to find a plausible
lacuna in the Cantorian proof that numbers are not timeless. But, what they
have found is more paradoxes akin to the Liar. Cantor’s paradox concerned the
set of all the

*other*sets because the set of all the sets would have had to contain itself as one of its own members, and we do not normally think of collections like that. But as Russell thought about Cantor’s counter-intuitive mathematics, he considered the collection of all the sets that do not belong to themselves: If that collection was a set, then it would belong to itself if, and only if, it did not belong to itself. That is basically Russell’s paradox. Like Cantor’s, it is not obviously paradoxical – it just means that there is no such set – but Russell thought of sets as the definite extensions of definite predicates, and predicate versions of his paradox are more obviously paradoxical. E.g. consider W.V.O. Quine’s version: ‘Is not true of itself’ is true of itself if, and only if, it is not true of itself. That is paradoxical because we naturally assume that ‘is not true of itself’ will either be true of itself, or else it will not. But if predicate expressions can be about as true as not of themselves, then it would follow from the meaning of ‘is not true of itself’ that insofar as ‘is not true of itself’ is true of itself it is not true of itself, and that insofar as it is not true of itself it is not the case that it is not true of itself. And it would follow that ‘is not true of itself’ is about as true as not of itself.
Russell also thought of
definite descriptions as names, and the English name for 111,777 – one hundred
and eleven thousand, seven hundred and seventy seven – has nineteen syllables.
According to Russell, 111,777 is the least integer not nameable in fewer than
nineteen syllables, and Berry’s paradox is that ‘the least integer not nameable
in fewer than nineteen syllables’ is a description of eighteen syllables.

^{2}Again, that is not very paradoxical; we can always use a false description as a name – cf. ‘Little John’ – and ‘John’ can name anything in one syllable. But consider the following two sentences.^{3}*The number denoted by ‘1’. The sum of the finite numbers denoted by these two sentences.*The first sentence denotes 1, so if the second sentence denotes anything, then it denotes a finite number, say*x*, where 1 +*x*=*x*, and there is no such number. So if the second sentence denotes anything, then it does not denote anything. But it cannot simply fail to denote, because if it does not denote anything, then the sum of the finite numbers denoted by those two sentences is 1. Since the second sentence denotes 1 if, and only if, it denotes nothing, perhaps it denotes 1 as much as not. Cf. how ‘King Arthur’s Round Table’ began as a definite description and ended up referring more vaguely.
In stark contrast, the
set-theoretic paradoxes – e.g. Cantor’s – do not have resolutions akin to the
present resolution of the Liar paradox: How could a collection of numbers be as
variable as not? (Collections of numbers are not like collections of noses, so
it could not be like Pinocchio’s nose.) Those paradoxes do have a fuzzy logical
resolution, via fuzzy sets, though. And those taking L to be true and false can
find it true and false that some collections belong to themselves. And those
taking natural Liar sentences to be nonsensical often have a formalist take on
infinite number. And of course, if the set-theoretic paradoxes have the same
underlying cause as the semantic paradoxes – as Russell thought – then they
should all be resolved in similar ways. But, if there are two kinds of paradox
here – as Ramsey thought – then the inability of the present approach to
resolve the set-theoretic paradoxes would hardly count against it. On the
contrary, that inability would amount to some evidence for it, by helping it to
explain the attractions of the major alternatives, especially the formal ones:
A non-classical logic would be very useful were one trying to fly in the face
of a mathematical proof.

Notes

1. For a detailed
history, see Ivor Grattan-Guinness,

*The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to GĂ¶del*(Princeton and Oxford: Princeton University Press, 2000).
2. Attributed to G.G.
Berry by Bertrand Russell, ‘Mathematical Logic as based on the Theory of
Types’,

*American Journal of Mathematics***30**(1908), 222–262.
3. Based on Keith
Simmons, ‘Reference and Paradox’, in J.C. Beall (ed.)

*Liars and Heaps: New Essays on Paradox*(Oxford: Clarendon Press, 2003), 230–252. Simmons’ version was more complicated, and omitted the crucial word ‘finite’.
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