Re: What is the intuitive meaning of Goedel's undecidable statement?
- From: "Charlie-Boo" <shymathguy@xxxxxxxxx>
- Date: 30 Apr 2006 16:12:57 -0700
Newberry wrote:
Is Godel's undecidable wff meaningless? It expresses "This is
unprovable." so it expresses " 'This is unprovable." is unprovable."
which expresses " ' "This is unprovable." is unprovable.' is
unprovable." etc.
Goedel's sentence is meaninless because it is analogical to
~Ex((Px & ~Px) & Qx)
Why this is meaningless can be seen from a similar example:
Ax((Px & ~Px) --> Qx)
It can be translated into English e.g. as "All square circles are
green." This cannot be pictured because square circle cannot be
pictured. What is the meaning of
~E(Px &~Px) =eqv= "There are no square circles"?
It has a metalinguistic interpretation: " 'Square circle' does not
denote anything." Wittgenstein, who was a proponent of the picture
theory of language, said that P & ~P was a limiting case (or something
to that effect.) What is the meaning of "P & ~P"? It is not a picture
of anything! ~(P & ~P) can be interpreted just like that: " 'P & ~P' is
not a picture of anything." But since "P & ~P" is not a picture of
anything you get meaningless formulas as soon as you start applying
logical operators to it.
An empty set is still a set.
C-B
As I said, occurrence relevance logic behaves exactly like this
although its author did not have this interpretation in mind.
.
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