Re: My investigations into Godels Incompleteness Theorem

John Jones says...

Thankyou for your well-received explanation which I read slowly and
understood. Can you say what happens next please?

That depends on whether you agree with the fixed point lemma,
or not. The route to Godel's incompleteness theorem has the
following parts:

1. Establish a correspondence between sentences (about arithmetic,
in the case of Godel, or about strings, in our variant) and objects
in the domain of discourse (natural numbers, in Godel's case, strings
in ours).

2. Establish the fixed point lemma. In Godel's case, for every formula
Phi(x) in the language of arithmetic, there is a sentence S in the
language of arithmetic such that S is true if and only if Phi holds
of the natural number corresponding to S. In our variant, for every
property P of strings, there is a sentence S such that S is true if
and only if the string corresponding to S has property P.

3. Establish that theoremhood is definable. In Godel's case, that
means coming up with a formula Provable(x) in the language of
arithmetic such that for any sentence S, S is provable from the
axioms of PA if and only if Phi holds of the natural number
corresponding to S. For our variant, we would find a property
"is provable" of strings such that for any sentence S, S is a
theorem if and only if the string corresponding to S has the
property "is provable".

4. Use the fixed point lemma to come up with a sentence S such that
S is true if and only if the natural number (or string, in our case)
corresponding to S satisfies the property of not being a the code of
a theorem.

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: add support for other languages
    ... language DLLs or a single DLL with multiple language support, ... Just compile in Unicode ... You can create a hand-edited resource that contains Unicode strings, but you have to do it ... I don't know where is the language support comes into ...
    (microsoft.public.vc.mfc)
  • Re: Two Questions about "strlen", "strcat" and "strcpy"
    ... >>Important is that we have in the standard language a way of using ... > No. zero terminated strings is the whole problem in the first place. ... > programmer to think in terms of implementation and constantly respin ... The standards comitee refuses any change, ...
    (comp.lang.c)
  • Re: GNU gettext
    ... The gettext documentation explains how the keys work. ... translation files, or that you'd need to write a small utility to help ... gettext was desined for plain C, the keys are C strings ... as a developer you have no clue what every language ...
    (microsoft.public.dotnet.languages.csharp)
  • Re: How come Ada isnt more popular?
    ... A language with a Hinldey-Milner type system ... fixed size strings, unbounded strings, suffix tree. ... container varies, the element does not. ...
    (comp.lang.ada)
  • Re: Substituting the main menu bar(s)
    ... I have tab-separated-value table of strings maintained with Excel, with a column for each language. ... All it does is replace the strings according to the translation table. ... I also have a batch file which calls the translation app, and then invokes the compiler on each resource-only DLL project. ...
    (microsoft.public.vc.mfc)