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

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Peter Percival

Posts: 1,304
Registered: 10/25/10
Re: Truth behind Paradox of Self Reference
Posted: Aug 25, 2014 3:25 PM
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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
> in adamantly insisting that Incompleteness is _only_ about the
> 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
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/21/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/21/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/21/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/21/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/22/14
Read Re: Truth behind Paradox of Self Reference
Antti Valmari
8/22/14
Read Re: Truth behind Paradox of Self Reference
namducnguyen
8/22/14
Read Re: Truth behind Paradox of Self Reference
namducnguyen
8/22/14
Read Re: Truth behind Paradox of Self Reference
Kaz Kylheku
8/22/14
Read Re: Truth behind Paradox of Self Reference
namducnguyen
8/22/14
Read Re: Truth behind Paradox of Self Reference
namducnguyen
8/22/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/22/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/22/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/22/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/22/14
Read Re: Truth behind Paradox of Self Reference
namducnguyen
8/22/14
Read Re: Truth behind Paradox of Self Reference
Kaz Kylheku
8/23/14
Read Re: Truth behind Paradox of Self Reference
namducnguyen
8/23/14
Read Re: Truth behind Paradox of Self Reference
namducnguyen
8/23/14
Read Re: Truth behind Paradox of Self Reference
namducnguyen
8/23/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/23/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/25/14
Read Re: Truth behind Paradox of Self Reference
Antti Valmari
8/25/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/25/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/25/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/25/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/25/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/26/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/26/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/26/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/26/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/26/14
Read Re: Truth behind Paradox of Self Reference
Kaz Kylheku
8/26/14
Read Re: Truth behind Paradox of Self Reference
Jim Burns
8/26/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/26/14
Read Re: Truth behind Paradox of Self Reference
Jim Burns
8/26/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/27/14
Read Re: Truth behind Paradox of Self Reference
Jim Burns
8/27/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/27/14
Read Re: Truth behind Paradox of Self Reference
Jim Burns
8/27/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott
8/26/14
Read Re: Truth behind Paradox of Self Reference
Kaz Kylheku
8/26/14
Read Re: Truth behind Paradox of Self Reference
Jim Burns
8/26/14
Read Re: Truth behind Paradox of Self Reference
Jim Burns
8/25/14
Read Re: Truth behind Paradox of Self Reference
namducnguyen
8/25/14
Read Re: Truth behind Paradox of Self Reference
namducnguyen
8/25/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/27/14
Read Re: Truth behind Paradox of Self Reference
Antti Valmari
8/27/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/27/14
Read Re: Truth behind Paradox of Self Reference
Newberry
9/3/14
Read Re: Truth behind Paradox of Self Reference
namducnguyen
9/3/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
9/3/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/23/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/23/14
Read Re: Truth behind Paradox of Self Reference
Jim Burns
8/23/14
Read Re: Truth behind Paradox of Self Reference
Peter Percival
8/26/14
Read Re: Truth behind Paradox of Self Reference
Rosario1903
8/23/14
Read Re: Truth behind Paradox of Self Reference
Peter Olcott

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