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.
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?
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The Two Dennises
Hannah Mortimer observes a close encounter of the same kind.
Heraclitus (c.500 BC)
Harry Keith lets flow a stream of ideas about permanence and change.
Does the Cosmos Have a Purpose?
Raymond Tallis argues intently against universal intention.
Is Driving Fossil-Fuelled Cars Immoral?
Rufus Duits asks when we can justify driving our carbon contributors.
Abelard & Carneades Yes & No
Frank Breslin says 'yes and no' to presenting both sides of an argument.
Frankl & Sartre in Search of Meaning
Georgia Arkell compares logotherapy and atheistic existentialism.
Luce Irigaray
Luce Irigaray, now ninety-two years old, was, among many other things, one of the most impactful feminists of the 1970s liberation movements - before she was marginalised, then ostracised, from the francophone intellectual sphere.
Significance
Ruben David Azevedo tells us why, in a limitless universe, we’re not insignificant.
The Present Is Not All There Is To Happiness
Rob Glacier says don’t just live in the now.
Philosophers Exploring The Good Life
Jim Mepham quests with philosophers to discover what makes a life good.