Re: Deep Thoughts # 17: Liar Paradox is a Formal Metamathematical Theorem

From: Josh Purinton (usenet-noreply.a.jp_at_xoxy.net)
Date: 11/22/04 

Date: Mon, 22 Nov 2004 05:18:00 GMT

In article <3df1e59f.0411212057.669a8568@posting.google.com>,
Charlie-Boo <chvol@aol.com> wrote:
>David C. Ullrich <ullrich@math.okstate.edu> wrote
>> he's not defining provability to be the
>> same as truth. (He's setting up a _specific_ formal system in
>> which the two coincide. There's a big difference.)
>
> Then what is he defining provability to be in his system?

Smullyan doesn't provide a separate definition for any system in the
book; rather, he uses the standard one. As he writes on page 1, "We
assume the reader to be familiar with the basic notions of first-order
logic -- [including] provability (in some complete system of first-order
logic with identity)".

> Among all systems that we can define with whatever formalism we are
> using, there is the set of them in which truth and provability
> coincide. If you say to define provability in a system to be the same
> as truth, then you are saying to assume that you are in one of the
> systems in this set.

You wrote: ``The sentence "It is true." expresses the provability
predicate, which in English is the truth predicate.''

What makes you believe that truth and provability coincide for English? 

-- 
Josh Purinton

Relevant Pages