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Topic: Truth behind Paradox of Self Reference
Replies: 58   Last Post: Sep 3, 2014 4:23 PM

 Messages: [ Previous | Next ]
 Peter Percival Posts: 2,623 Registered: 10/25/10
Re: Truth behind Paradox of Self Reference
Posted: Aug 25, 2014 3:25 PM

Nam Nguyen wrote:

> [...] this is a requirement
> in Gödel's paper that T should be about the concept of the natural
> numbers.

I don't think that's so. Second order predicate calculus and the set
theories ZF and NBG are both incomplete. (They are both incomplete in
both senses: there is a formula phi (different for the different
theories) such that neither phi nor ~phi is provable _and_ there is a
formula (ditto) that is both true and unprovable. None of of those
three theories is about the natural numbers. (The phrase 'the concept
of' is just so much noise.)

> [...]
>
> The key point being is some of my opponents were (and still are) wrong
> un-provability (undecidability)in T (Gödel'sP) of Gödel's sentence
> G and not about the truth value of G in the natural numbers.

Your emphasised 'only' is a red herring. Here are two facts.
i) Gödel exhibited a formula, call it G, such that P neither proves G,
nor does it prove ~G.
ii) The G in question is true.
Imagine that Gödel had written his paper without the introductory chat.
Imagine, further, that he kept quiet thereafter. Then all one could
attribute to Gödel would be i); but that would not stop some other
person showing ii). Incompleteness in sense i) (as proved (more than
once, as it happens) by Gödel) is a syntactic matter syntactically
proved. The further _interpretation_ ii) is a different matter.
>
> Under the assumptions T is about the natural numbers and is consistent,
> _G being true in the natural numbers and G isn't provable in T are_
> _materially and logically equivalent_ in his paper: one can _NEVER_
> find G being unprovable (undecidable) in T and yet being _false_ !

When you write "unprovable (undecidable)" does that indicate that you
think the two concepts are the same? They are not. 0=/=S0 is
unprovable (assuming consistent) but not undecidable. Your last
paragraph is either false or much in need of clarification.

--
[Dancing is] a perpendicular expression of a horizontal desire.
G.B. Shaw quoted in /New Statesman/, 23 March 1962

Date Subject Author
8/20/14 Peter Olcott
8/21/14 Peter Percival
8/21/14 Peter Olcott
8/21/14 Peter Percival
8/21/14 Peter Olcott
8/22/14 Antti Valmari
8/22/14 namducnguyen
8/22/14 namducnguyen
8/22/14 Kaz Kylheku
8/22/14 namducnguyen
8/22/14 namducnguyen
8/22/14 Peter Percival
8/22/14 Peter Percival
8/22/14 Peter Percival
8/22/14 Peter Percival
8/22/14 namducnguyen
8/22/14 Kaz Kylheku
8/23/14 namducnguyen
8/23/14 namducnguyen
8/23/14 namducnguyen
8/23/14 Peter Percival
8/23/14 Peter Percival
8/25/14 Antti Valmari
8/25/14 Peter Olcott
8/25/14 Peter Percival
8/25/14 Peter Olcott
8/25/14 Peter Percival
8/25/14 Peter Olcott
8/26/14 Peter Percival
8/26/14 Peter Olcott
8/26/14 Peter Percival
8/26/14 Peter Olcott
8/26/14 Kaz Kylheku
8/26/14 Jim Burns
8/26/14 Peter Olcott
8/26/14 Jim Burns
8/26/14 Peter Olcott
8/27/14 Jim Burns
8/27/14 Peter Olcott
8/27/14 Jim Burns
8/27/14 Peter Olcott
8/26/14 Kaz Kylheku
8/26/14 Jim Burns
8/26/14 Jim Burns
8/25/14 namducnguyen
8/25/14 namducnguyen
8/25/14 Peter Percival
8/27/14 Antti Valmari
8/27/14 Peter Percival
8/27/14 Newberry
9/3/14 namducnguyen
9/3/14 Peter Percival
9/3/14 Peter Percival
8/23/14 Peter Percival
8/23/14 Jim Burns
8/23/14 Peter Percival
8/26/14 Rosario1903
8/23/14 Peter Olcott