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

From: Charlie-Boo (chvol_at_aol.com)
Date: 01/06/05 

Date: 6 Jan 2005 02:31:28 -0800

Daryl McCullough wrote:
> Charlie-Boo says...
>
> >How else could you define system N so that provability and truth
> >coincide?
>
> I don't know what your point is.

Do you have to know the purpose of a question in order to determine how
you will answer it? Isn't truth independent of one's perceived
purpose?

> My point is that Smullyan did not define
> *provability* in terms of truth, he defined the *axioms* of system N
in terms of
> truth. Provability (as usual) is defined in terms of logical
deduction.

My answer follows each question in (parentheses):

1. What is the purpose of an Axiomatic System? (To create a system of
proof in which all provable statements are all true statements)

2. What did Leibniz and Hilbert attempt? (They attempted to construct
such an axiomatic system for all of Mathematics.)

3. Does standard Logic require that provability coincide with truth?
(No)

4. If A is defined in terms of B and B is defined in terms of C, is A
defined in terms of C? (Yes)

5. Is the Mathematical significance of a definition a function of the
person who conceived of, wrote, explained or advocated that definition?
(No)

6. Is it mathematically possible to deduce the true sentences using
deduction rules? (No)

7. Does it make sense to say that two Mathematicians can use the same
system, but one of them is wrong because he used a different wording in
his definition of the system? (No)

8. Where and how are inference rules used in System N? (Nowhere)

9. How else could Smullyan have defined system N and assured that
provability coincides with truth? (There is no other way.)

10. By what principles of Mathematics is it wrong to define an
axiomatic system in which provability is defined to be truth? (There
are none.)

With which of my 10 answers do you disagree?
C-B

> --
> Daryl McCullough
> Ithaca, NY

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