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

From: Charlie-Boo (chvol_at_aol.com)
Date: 01/03/05 

Date: 3 Jan 2005 12:06:47 -0800

Daryl McCullough wrote:
> Charlie-Boo says...
>
> >How else can you arrange for provability to coincide with truth in
> >System N, if not by letting the axioms be the true sentences and the
> >rules, in effect, be inconsequential?
>
> The rules aren't inconsequential. The rules for Smullyan's system are
the usual
> rules for first-order logic.

Ok, then consider the following two systems:

1. System N as defined by Smullyan.

2. System N' which is System N minus the rules of inference, i.e., it
simply has no rules of inference.

Now, what is the consequence of having the rules in N vs. no rules in
N' i.e. what is the difference between N and N'?

Answer: None. They have the same theorems.

How else could you define system N so that provability and truth
coincide?

C-B

> --
> Daryl McCullough
> Ithaca, NY

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