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Paul Tissier argues that Russell’s Paradox isn’t really a paradox.
Bev and Ali are deep in discussion.
Ali: There can’t be real paradoxes, can there? Those that purport to be paradoxes actually contain ambiguities in meaning, neglected information, hidden assumptions… In short, it’s all smoke and mirrors.
Bev: You might think so. However, Russell’s Paradox in set theory has attracted a lot of serious attention in the literature. In fact, Gottlob Frege, a leading logician and mathematician at the time, felt that this paradox devastated his fundamental work on set theory and the foundations of mathematics.
Bertrand Russell by Stephen Lahey
Ali: So if Russell’s Paradox holds up, it would, rather worryingly, shake the foundations of mathematics?
Bev: Right. So let’s see what is involved. We all know – or think we know – what a ‘set’ is: a collection of all those and only those entities with a defined property. A set’s members can be simple objects, or can be sets themselves. It’s the last possibility we’re interested in here. Sets can also be non-self-membered, or self-membered: they can belong to themselves, or not. Examples of the former would include the set of all finite sets, which is infinite and hence non-self-membered. On the other hand, the set of all infinite sets is itself infinite and so must be included in itself, so it is self-membered. All sets either are non-self-membered or self-membered.
Ali: OK… But I feel like I’m about to be trapped, or perhaps conned.
Bev: Well consider the set of all non-self-membered sets, which for the sake of argument, let’s call ‘N’.
Ali: Hang on! Does such a set even exist?
Bev: Why not? Just as with all other sets, membership in the set is clearly defined, and a set is defined by its members.
Ali: OK. Fine…
Bev: So we can ask – as Bertrand Russell himself did, no doubt with an impish grin – is the set N self-membered or non-self-membered? Is the set of all non-self-membered sets a member of itself, or not? If the set of all non-self-membered sets is self-membered then it is non-self-membered; yet if it is non-self-membered then it is self-membered! See? A blatant paradox.
Ali: (after a long pause) So what’s to be done about it?
Bev: There have been various suggestions. For example, that set N simply doesn’t exist; or is too large, too unruly to be called a set, so we should call it something else, perhaps a ‘class’; or, members of a set are a different kind of thing to the set containing them, so that no set can be self-membered.
Ali: The last alternative seems to me to be nearest the mark. After all, if we imagine a set to be constructed from members of the set, this suggests that these members are somehow logically prior to the set to which they belong. If so, then it makes no sense to say one of the members of the set is the set itself – the very set which is in the process of being constructed! This lack of sense shows itself if we ask, ‘‘If a set ‘S’ is a member of itself, what exactly is ‘itself’ referring to?’’ S without S included would mean S is non-self-membered, which is contrary to our definition. Yet S with S included is not just S, but S with an addition. In neither case do we have S being self-membered. Thus no set can be self-membered. Russell’s paradox disappears.
Bev: But against that, one could argue that there is no logical priority necessary for the construction of a set, since there is no ‘construction’. A set either exists as defined, or it does not. So if set S is a member of itself, then as we go through the members of S, we find S itself included. It has not had to be added in. It is simply there by the same token that all the other members of the set are there.
Ali: So the set of all non-self-membered sets might be self-membered, or it might not be?
Bev: I think we can make a more telling argument. Two points need to be stressed concerning the properties ‘self-membered’ and ‘non-self-membered’. Firstly, the properties are derivative – they cannot stand alone, as ‘being finite’ or ‘being infinite’ can. They come into existence solely due to prior existent set properties. Secondly, and encompassing the first point, being ‘self-membered’ or ‘non-self-membered’ must be the result of a definite decision procedure: the ascription must be true or false, be logically possible, ascertainable. In light of this, how could the question ‘Is N self-membered or non-self-membered?’ have an answer? It couldn’t. To give an answer is to answer an unanswerable question, and that’s a contradiction. And of course, it would be quite absurd to suggest that N is self-membered because it is non-self-membered – and vice versa.
Ali: The same seems to apply to N* – the set of all self-membered sets, a kind of symmetrical partner to N. So in the case of N*, if we say that N* is self-membered, or indeed non-self-membered, for no other reason than it is self-membered, or non-self-membered, then we are saying nothing at all. The same goes for N, despite the fact that it appears paradoxical.
Bev: Right. Russell’s Paradox is not a paradox. The apparent paradox is merely the result of following through a form of words which purport to describe an actuality; but actually nothing is picked out from mathematical reality by the phrase ‘the set of all sets that are not members of themselves being self-membered or non-self-membered’.
Ali: The statement then that all sets are self-membered or non-self-membered is clearly wrong in the case of N and N*. Both are neither non-self-membered nor self-membered.
Bev: And this should not be surprising. Perhaps we have a parallel with the square root of -1. It is a perfectly serviceable number, yet neither greater than or equal to zero, or less than zero.
© Paul Tissier 2022
Paul Tissier is a Physics and Maths tutor at Brighton College, East Sussex.