The Set-theoretical Paradoxes

I’ve been writing about the semantic paradoxes (see recent posts and my homepage), so perhaps I should say something about the set-theoretical paradoxes. The seminal paradox of Bertrand Russell (1902: ‘Letter to Frege’, in 1967: Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, 124–5), for example, concerns the class of classes that do not belong to themselves. Classes are extensions of predicates: all and only the things that satisfy the predicate belong to the class. Now, some classes – e.g. the class of humans – do not belong to themselves – the class of humans is a class, not a human – and the class of all such classes is paradoxical: it belongs to itself if, and only if, it does not. Russell conceived this paradox when thinking about the set-theoretical paradoxes, because a class is a kind of set; but, Russell’s paradox can also be expressed directly in terms of predicates:

Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. (Ibid, 125)

Note that “is a human” is a predicate expression, not a human, so it does not describe itself; the question is, what about “does not describe itself”? It describes itself if, and only if, it does not, which is paradoxical (it is widely known as Grelling’s paradox). But, we might as well say that “does not describe itself” describes itself insofar as it does not, from which it follows that it describes itself as much as it does not. So the question arises, could the class of classes that do not belong to themselves belong to itself as much as not? Classes can be like that, e.g. the class of all men would be like that if some hominids had been about as human as not. If we call one such hominid ‘Strider’, then it was about as true as not that Strider was human. Were there no such hominids, then some human would have had non-human parents.
......That – the fuzzy set – resolution of Russell’s paradox coheres rather well with my preferred resolutions of the semantic paradoxes. But, the set-theoretical paradoxes originally arose from the mathematics of Georg Cantor, and they concerned, not classes of classes, but numbers of numbers. And of course, numbers are far from fuzzy.
......You and I, for example, are two people, and there is surely no doubt that we know what is meant by ‘two’, for all the uncertainty over the personhood of Strider. So, let us begin with 0, 1, 2, 3 and so forth, the products of the process of adding 1 to the previous number, starting with 0. Rather trivially, the collection of all those numbers is all of them, referred to collectively; and while some collections – e.g. stamp collections – are variable, if a collection is non-variable then we can say that it is a set (as in “a set of stamps”). On this conception, a set is some particular number of logical objects. To include the numbers 0 and 1, making this conception more like the standard conception (and simplifying proofs), let us also include logical objects that can play the role of singletons – sets with a single element – and Ø, the empty set. So, given that we have a set {0, 1, 2 …}, it contains some definite number of numbers, say א (aleph) of them.
......Cantor showed that every set has more subsets than it has elements, in the cardinal sense of ‘more’. (Two sets have the same cardinal number of elements when the elements of each set can all be paired up with those of the other, cf. Hume's principle) Let S be any set, and let P (for ‘powerset’) be the set of all its subsets (including Ø and S). If S and P had the same cardinality, then there would be one-to-one mappings from S onto all of P, so let M be one such mapping. Let a subset of S, say D, be specified as follows: For each member of S, if the subset that M maps it to contains it, then D does not contain it, and otherwise D does. The problem is that since D differs from every subset that M maps the members of S to, D differs from every subset of S, whereas D is by definition a subset of S. So, D is contradictory, and so there is no such M. So S and P do not have the same cardinality, and since P contains a singleton for each element of S, P is bigger than S. So, {0, 1, 2 …} has beth-one subsets, where beth-one is bigger than aleph, and the set of all those sets has beth-two subsets, and so on.
......If that endless sequence of bigger and bigger sets is a non-variable sequence, then there is a union – a set of the elements – of all those sets, which is even bigger, with בω (beth-omega) sets. (Omega is the ordinal number of the sequence 1, 2, 3 and so forth.) And that union has בω + 1 subsets, and so on: for any such set there is the set of its subsets, and for any endless sequence of such sets there is, if it is a non-variable sequence, its union. In total, there is a sequence of sets – and a corresponding sequence of numbers, the sizes of those sets – which must be variable; were it not, we would have moved on from that ordered set of sets to its union, and thence to the subsets of that union (and so on). But of course, it is paradoxical that our total sequence of numbers is variable – is of necessity growing forever – because few of us think that numbers that do not already exist could suddenly appear. Suppose, for example, that the number 101 had not always existed; would that not mean that there was once a time when there were no such possibilities as, for example, the possibility of 101 Dalmatians? And note that this paradox cannot be resolved as Russell’s paradox was resolved above, because the idea of something being as variable as not is nonsensical.
......Nevertheless, the intuition that numbers are atemporal is not unquestionable, because new possibilities can be constructed out of more general possibilities. You were always possible, for example, and yet the possibility of you in particular was only distinct from the more general possibility of people just like you once you existed (to be directly referred to). And it is not too odd to think of arithmetic as constructed from such logical concepts as those of possibility and class. E.g. the obvious meaning of “2 + 2 = 4” is that if we had two things of some kind, then if we got another two of that kind we would have four. So, it is conceivable that, while 101 Dalmatians were always possible, there was once a time when that possibility only existed as part of a more general possibility (of bigger numbers). Such constructivism can be defended atheistically – e.g. see George Lakoff and Rafael E. Núñez (2000: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, New York: Basic Books) – and theistically, e.g. see Paul Copan and William Lane Craig (2004: Creation out of Nothing: A Biblical, Philosophical, and Scientific Exploration, Grand Rapids, MI: Baker Academic).
......Still, whether we are atheists who believe that the human brain evolved in a finite world, or theists who entertain divine ineffability and infinitude, we would have such reasons to doubt that we could ever be justifiably sure about the nature of infinity. And another reason why we should keep an open mind is that, while it is clearly counter-intuitive to think of the finite cardinal numbers as temporal, it is, if you think about it, no less counter-intuitive to think of them as atemporal. E.g. the arithmetic of such numbers as א and ω is very different to that of the finite cardinal numbers, whence the theoretical behaviour of that many objects is counter-intuitive. Hilbert’s famous Hotel can be built upon Galileo’s paradox, for example. And the difference between cardinal and ordinal arithmetic gives rise to the counter-intuitive behaviour of my quasi-supertask (2003: ‘Infinite Sequences, Finitist Consequence’, British Journal for the Philosophy of Science 54, 591–9). And the infinite set of the finite cardinal numbers covers the whole range of the finite (in units), and yet every one of those numbers is infinitely far from infinite, whence Lévy’s paradox. For more examples, see José Benardete (1964: Infinity: An Essay in Metaphysics, Oxford: Clarendon Press), and Peter Fletcher (2007: ‘Infinity’, in Dale Jacquette, Philosophy of Logic, Amsterdam: Elsevier, 523–585).
......Intuitively, numbers are timeless; but while it is certainly possible that there is a set of all the finite cardinal numbers, it is also possible that there is not. Both possibilities are counter-intuitive, so both can be supported in ways that would seem compelling were it not for that ‘both’. So, one might think that modern mathematics would have been based on results that follow, not just from one, but from both possibilities. However, such is not the case. Now, the ubiquity of the standard real number line might be explained by its being easy to use, simple and familiar, but there is a similar bias towards assuming that there is a non-variable collection {0, 1, 2 …} in such fundamental research areas as theoretical physics and pure mathematics, which is a little puzzling. For clues, see Ivor Grattan-Guinness (2000: The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel, Princeton University Press), and Peter Markie (2013: ‘Rationalism vs. Empiricism’, in Edward N. Zalta, The Stanford Encyclopedia of Philosophy).
......It might be objected that {0, 1, 2 …} is not an informal set in pure mathematics, but is an axiomatic set defined by means of a formal logic. But that would be to ignore, not to explain, the mystery. We know perfectly well what the cardinal numbers 0, 1, 2, 3 and so forth are; if we had some axioms that did not describe them, we would not throw those numbers away and start using those axioms instead, however nice their formal properties were. To do so would hardly be scientific.
......Perhaps I should add that we do not get a third kind of set-theoretical paradox from the axiomatic conception. Paradoxes arise when we have contradictory beliefs, and formal structures have no intrinsic meaning; formal axiomatic sets only give us mathematical models of set-theoretical paradoxes. So, while it is true that paradoxes can be avoided if we use formal sets, we did not really resolve the set-theoretical paradoxes by moving from naïve set theory to axiomatic set theory.

On a formal Liar

A lot of formal work is done on the Liar paradox, so note that a formal logic is no more than a mathematical model of correct reasoning.
......E.g. Richard Heck was tempted by his formal version of the Liar paradox (Thought 1(1): 36–40) ‘to conclude that there can be no truly satisfying, consistent resolution of the Liar paradox’ (p. 39). And he did have a strong model because he assumed little more than two very weak logical principles, his equations 3 and 4 (p. 38). But it was still only a model.
......Heck’s informal illustration of equation 3 was: ‘It cannot be both that snow is white and that “snow is not white” is true’ (p. 38). That is unobjectionable because ‘snow is not white’ just means that it is not the case that snow is white. Insofar as snow is white, the claim made by ‘snow is not white’ is not true. And equation 4 was similar, e.g. it cannot be both that snow is not blue and that ‘snow is not blue’ is not true.
......Heck had a model of the Liar paradox because he had already introduced a term, λ, defined by his equation 2 (p. 36), which was a formal version of such definitions as the following: ‘L’ names the self-referential claim made by ‘L is not true’. It follows from that informal definition that insofar as L is true, it is true that L is not true. And L should conform to the logic behind equation 3, so insofar as L is true, it is not true that L is not true. But it does not follow logically that L is not true, because L may well be as true as not.
......Formally, equations 2 and 3 rule out T(λ). And similarly, equations 2 and 4 rule out ¬T(λ). But for a solution to Heck’s problem to be truly satisfying, it need only stay true to the underlying purpose of one’s formal logic. If we want to include terms like λ in our formal language, then we will need a better model of truth than T, but when deducing theorems from axioms we won’t normally need to allow for the possibility of terms like λ. And for such purposes as A.I. paraconsistent logic is sufficiently consistent.

The no-no Paradox

Roy Sorensen (2001: Vagueness and Contradiction, Oxford: Clarendon Press, 169) considered the following pair of sentences:
......The neighbouring italicized sentence is not true.
......The neighbouring italicized sentence is not true.
While it is logically possible that one of those sentences – or rather, one of the claims made by those sentences – is true and the other false, those two tokens of that sentence-type should have the same truth-value because there are no significant contextual differences between them. It is therefore plausible that each is as true as not. For each token, were the claim expressed by it as true as not, the other claim would be as untrue as not, which clearly coheres with it too being as true as not.

The Unsatisfied Paradox

In this month's issue of The Reasoner (page 185), Peter Eldridge-Smith gave the following informal description of his Unsatisfied paradox:

My favourite predicate just happens to be 'does not satisfy my favourite predicate'. Crete satisfies 'does not satisfy my favourite predicate' iff Crete does not satisfy my favourite predicate. Therefore, Crete satisfies my favourite predicate iff Crete does not satisfy my favourite predicate.

And not just Crete, there is no thing that satisfies Peter's favourite predicate, and no thing that fails to satisfy it without it also not being the case that it fails to satisfy it. Nevertheless, Peter's favourite predicate could be as true as not of Crete, or anything else. Predicates can do that, e.g. 'is blue' is as true as not of an object that is as blue as not, and some predicates apply equally to all things, e.g. 'is a thing' is true of all things.

The Pinocchio paradox

Suppose that Pinocchio's nose grows if, and only if, he says something that is not true, and that he says "My nose is growing". Then his nose is growing if and only if it is not growing. (This paradox originated with Veronique Eldridge-Smith.) According to Peter Eldridge-Smith:

The Pinocchio scenario is not going to arise in our world, so it is not a pragmatic issue. It seems though that there could be a logically possible world in which Pinocchio’s nose grows if and only if he is saying something not true. However, there cannot be such a logically possible world wherein he makes the statement ‘My nose is growing’.

In the world in which Pinocchio's nose grows and shrinks in such a way, suppose that he says, of various uniformly coloured objects, that they are blue. What happens if the object is as blue as not? (There must be such colours, because otherwise some colour that is blue is the same colour as some colour that is not blue.) Well, whatever happens, that could also be what happens when he says "My nose is growing". It is, for example, possible that Pinocchio's nose is in a quantum-mechanically entangled state, as much growing as not. That seems to be a logically possible world.

Curry's paradox

Consider C, which says that if C then B, where B says that black is white. Suppose that C is true. Then it is true that C implies B. So if C is true, then B is true. But therefore C is true. So B is true. That in a nutshell is Curry's paradox. Every step in that argument that black is white appears to be a logical step.
......But, ordinary modal logic operates in a space of descriptions that are either true or else false. So perhaps we should not have supposed, to begin with, that C is true, because C is true only insofar as it is not true (because insofar as it is true we get a contradiction) and so it is as true as not. That is a consistent possibility because it is as true as not that a contradiction follows from a statement that is as true as not (since it would follow from a statement that was false). And it resolves the paradox because ordinary logic breaks down with propositions that are only as true as not (e.g. see the Sorites paradox, higher-order vagueness, and the revenge problem for this resolution of the Liar paradox).
......So, maybe Curry's paradox is showing us that we should not even suppose the truth of some conditionals that are as true as not. Of course, C does just that, with its "if C". But that just means that its grammatical structure might be less logical than that of, say, A, the claim that A is not true (which basically says that if A is true then pigs fly). That might explain how all the steps of our fallacious argument could have been logical, while the Liar paradox needs us to presuppose bivalency.

Revenge revisited

The obvious resolution of the Liar paradox is that such self-descriptions as “this claim is not true” are as true as not. That resolution is obvious because it follows immediately from the problem, which is that insofar as they are not true, such claims are true. That is how the problem should be expressed, rather than as paradoxical biconditionals, if truth might be a matter of degree. And if a uniformly coloured object is as blue as not, then prima facie “the object is blue” is as true as not, so that is a prima facie logical possibility. Indeed, since the Liar paradox is a paradox, its less obvious resolutions are all very implausible, and so it is itself an indication that truth is in general a matter of degree.
......But, there is a fly in the ointment. What about “this claim is not even as true as not”? Were that claim as true as not, it would be false that it was not even as true as not. That is the revenge problem for this resolution. The problem is that insofar as the revenge claim is at least as true as not, it is false. But it follows that, while it is a bit more false than true, it is still about as true as not. So where is the problem? The revenge claim is almost as true as not. Now, if we use “vaguely true” to mean about as true as not, then we have another revenge claim, “this claim is not even vaguely true”. Insofar as that claim is vaguely true, it is false. It follows that it is as false as it is vaguely true, though. So this claim is just a bit more vaguely true. Again, where is the fly?
......It is trapped in the bottle of ointment, loudly buzzing. The buzz is that it is clearly not the case that “this claim is not true” is true. Were it true, it would be as it says it is, it would not be true. So it is not true. Even if it is as true as not, it is still not true enough to count as simply true. And furthermore, if “this claim is not even vaguely true” is vaguely true, then it what it claims is simply not true. How could it help to talk of the more vaguely true when “vaguely true” is such a vague term? Or so the buzzing goes. But, it is precisely because “vaguely true” is vague that there is no contradiction. The buzzing is understandable, but it does not spoil the ointment. It is understandable because we tend to imagine that things are black-and-white when we try to think logically.
......Consider the uniformly coloured object that is, in some contexts, as blue as not. Of course, blue is not normally a matter of degree. Oxford blue is deeper, not bluer, than Cambridge blue. “Oxford blue is blue” is not truer than “Cambridge blue is blue”. But, as greenish blue shades into bluish green there is bound to be a colour that is about as blue as not. There would otherwise be a blue colour that was the same colour as a colour that was not blue. And of course, a blue-green object will in some contexts look blue, e.g. when surrounded by red balls. But our object is in a context where it is as blue as not, so it does not look very blue. So I do not want to say that it is blue. If anything, I want to say that it is not the case that it is blue. But tempted as I am to say that it is not blue, the fact is that “the object is not blue” is, like “the object is blue”, only as true as not.

Simmons' Paradox

Keith Simmons told this story about ten years ago:

Suppose I’ve just passed by a colleague’s office, and I see denoting phrases on the board there. That puts me in the mood to write denoting phrases of my own, and so I enter an adjacent room, and write on the board the following expressions:

......pi
......six
......the sum of the numbers denoted by expressions on the board in room 213.

Now I am in fact in room 213, though I believe that room 213 is my colleague's office. I set you the task of providing the denotations of these expressions.

The question is, what is X when X = pi + 6 + X, and the obvious answer is “infinity”. Possible answers are omega, aleph-null, aleph-one, and so forth. There are lots of possible answers, so the “the” at the start of the third expression is a little deceptive. It is like asking for the name of the king of France; the obvious reply is, which king of France?
......But, what if “numbers” in the third expression was replaced with “finite numbers”? The expression “the present king of France” denotes nobody. But, if the third expression with “finite” does not denote anything, then the sum of the numbers denoted by expressions on the board would be pi + 6, so the third expression ought to denote pi + 6. And then it should denote twice that, whence it should denote thrice, and so forth. If it denotes anything, it does it inconsistently. So, it denotes nothing consistently if it denotes anything, but if it denotes nothing then it denotes pi + 6.
......The light on the horizon is that reference is in general a matter of degree. To why, imagine a man staggering through a desert and seeing a mirage, which he takes to be a pool. Coincidentally there is a pool just where he takes one to be, but it is obscured from his view by the mirage. As he staggers on he is constantly thinking “that pool looks cool” rather obsessively. And as he nears the pool its image gradually replaces the illusory one without him noticing. The referent of “that pool” gradually changes to the pool, and so he will at some point have referred to the pool only as much as not.
......What did “that pool” denote when the man was referring to the pool only as much as not? I would say that it denoted the pool, but only vaguely. And similarly, the third expression with “finite” refers us to pi + 6 insofar as it does not, so it refers us to pi + 6 as much as not. It denotes pi + 6, but only vaguely.

Why was the Big Bang not a Black Hole?

A documentary about what happened before the Big Bang was repeated on the BBC last night, and it got me wondering why the Big Bang was not a Black Hole, a Super-duper-massive Black Hole. There was all this matter, all the matter in the universe, in this tiny, tiny space. So why the explosion? And why an inflationary explosion? And why is the universe still accelerating? Did dark energy make it all happen? That does sound like a physics of the gaps.
......I had already been wondering why an amount of antimatter equal to the observable matter of the universe would not be in the form of an uncollapsed standing wave (like electron shells around atomic nuclei). The popular theory of where all the antimatter went is that there was originally a lot more extra matter and an equal amount of antimatter which annihilated each other. But that would just create a lot of heat and light, none of which could escape a Black Hole. But, were the antimatter in a standing wave, then the uncollapsed antimatter suffusing the primordial atom would make it effectively massless, so there would be no Black Hole, while the repulsive force between the matter and the antimatter would cause an explosive expansion. Furthermore, the appearance of dark matter would be explained, while the standing wave would enforce a certain uniformity, much as the inflationary period is supposed to have done.
......I have not heard of any such theory, so that thought is not even philosophy of physics, but listening to the physicists in that documentary made me wonder whether there might be such a theory. The things they were saying were pretty off the wall (according to each other).

Finished

Eternity Revisited