[FOM] A Question

[Alasdair Urquhart]

I think there is a problem with the definition of the implication
predicate.  If it's to be understood as a definition of
a predicate, it has to be written as something like:

 	impl([x], [y]) < - >  (x -> y),

where x and y are free variables.  But this is not a formula
of first order arithmetic, so the attempted definition fails.

Alasdair Urquhart


> Let L be a language capable of expressing its own syntax.  Add to L a
> two place predicate 'impl' satisfying
>
> (1) impl ([A], [B]) < - > (A - > B),
>
> where terms such as [A] are standard names of sentences A or of their
> Godel numbers.
>
> Let p be an arbitrary sentence of the object language.  Then impl(v,
> [p]) is a formula in one free variable.  By Godel's diagonalization
> lemma, it follows that there is a sentence W such that
>
> (2) W < - > impl ([W], [p]).
>
> From here on the argument for Curry's paradox (using (1) and (2)) can
> be used to deduce p.
 	Harry Deutsch