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5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / d. Richard's paradox

[problem with defining a closed set of real numbers]

2 ideas
Richard's puzzle uses the notion of 'definition' - but that cannot be defined [Russell]
     Full Idea: In Richard's puzzle, we use the notion of 'definition', and this, oddly enough, is not definable, and is indeed not a definite notion at all.
     From: Bertrand Russell (On 'Insolubilia' and their solution [1906], p.209)
     A reaction: The background for this claim is his type theory, which renders certain forms of circular reference meaningless.
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]