10 ideas
13373 | Typically, paradoxes are dealt with by dividing them into two groups, but the division is wrong [Priest,G] |
Full Idea: A natural principle is the same kind of paradox will have the same kind of solution. Standardly Ramsey's first group are solved by denying the existence of some totality, and the second group are less clear. But denial of the groups sink both. | |
From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §5) | |
A reaction: [compressed] This sums up the argument of Priest's paper, which is that it is Ramsey's division into two kinds (see Idea 13334) which is preventing us from getting to grips with the paradoxes. Priest, notoriously, just lives with them. |
13368 | The 'least indefinable ordinal' is defined by that very phrase [Priest,G] |
Full Idea: König: there are indefinable ordinals, and the least indefinable ordinal has just been defined in that very phrase. (Recall that something is definable iff there is a (non-indexical) noun-phrase that refers to it). | |
From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §3) | |
A reaction: Priest makes great subsequent use of this one, but it feels like a card trick. 'Everything indefinable has now been defined' (by the subject of this sentence)? König, of course, does manage to pick out one particular object. |
13370 | 'x is a natural number definable in less than 19 words' leads to contradiction [Priest,G] |
Full Idea: Berry: if we take 'x is a natural number definable in less than 19 words', we can generate a number which is and is not one of these numbers. | |
From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §3) | |
A reaction: [not enough space to spell this one out in full] |
13369 | By diagonalization we can define a real number that isn't in the definable set of reals [Priest,G] |
Full Idea: Richard: φ(x) is 'x is a definable real number between 0 and 1' and ψ(x) is 'x is definable'. We can define a real by diagonalization so that it is not in x. It is and isn't in the set of reals. | |
From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §3) | |
A reaction: [this isn't fully clear here because it is compressed] |
13366 | The least ordinal greater than the set of all ordinals is both one of them and not one of them [Priest,G] |
Full Idea: Burali-Forti: φ(x) is 'x is an ordinal', and so w is the set of all ordinals, On; δ(x) is the least ordinal greater than every member of x (abbreviation: log(x)). The contradiction is that log(On)∈On and log(On)∉On. | |
From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §2) |
13367 | The next set up in the hierarchy of sets seems to be both a member and not a member of it [Priest,G] |
Full Idea: Mirimanoff: φ(x) is 'x is well founded', so that w is the cumulative hierarchy of sets, V; &delta(x) is just the power set of x, P(x). If x⊆V, then V∈V and V∉V, since δ(V) is just V itself. | |
From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §2) |
13371 | If you know that a sentence is not one of the known sentences, you know its truth [Priest,G] |
Full Idea: In the family of the Liar is the Knower Paradox, where φ(x) is 'x is known to be true', and there is a set of known things, Kn. By knowing a sentence is not in the known sentences, you know its truth. | |
From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §4) | |
A reaction: [mostly my wording] |
13372 | There are Liar Pairs, and Liar Chains, which fit the same pattern as the basic Liar [Priest,G] |
Full Idea: There are liar chains which fit the pattern of Transcendence and Closure, as can be seen with the simplest case of the Liar Pair. | |
From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §4) | |
A reaction: [Priest gives full details] Priest's idea is that Closure is when a set is announced as complete, and Transcendence is when the set is forced to expand. He claims that the two keep coming into conflict. |
17914 | He made a molten sea, which was ten cubits across, and thirty cubits round the edge [Anon (Kings)] |
Full Idea: And he made a molten sea, ten cubits from the one brim to the other; it was round all about, and his height was five cubits: and a line of cubits did compass it round about. | |
From: Anon (Kings) (11: Book of Kings 1 [c.550 BCE], 7:23) | |
A reaction: In the sixth century BCE, this appears to give 3 as the value of Pi, though perhaps it shouldn't be taken too literally! |
19216 | Propositions (such as 'that dog is barking') only exist if their items exist [Williamson] |
Full Idea: A proposition about an item exists only if that item exists... how could something be the proposition that that dog is barking in circumstances in which that dog does not exist? | |
From: Timothy Williamson (Necessary Existents [2002], p.240), quoted by Trenton Merricks - Propositions | |
A reaction: This is a view of propositions I can't make sense of. If I'm under an illusion that there is a dog barking nearby, when there isn't one, can I not say 'that dog is barking'? If I haven't expressed a proposition, what have I done? |