Re: My investigations into Godels Incompleteness Theorem


Daryl McCullough wrote:
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

I think that these are the steps that indicate where my suspicions
might lie, if indeed they are grounded. Obviously this means some work
for you, but I have done the same for others on other topics and find
that it is also personally useful, especially in tidying up ones
understanding or presenting teaching material.

I did not want to say whether I went along with it or not at this
juncture, simply because if we became stuck on a particular point I
would not find out if it was central to the argument unless and until I
saw the argument develop in spite of the glitch.

But my concerns are too strong at the moment. I am assuming that these
concerns are central to what I claimed was the flaw in Godels theorem.
This assumption might be wrong, and I run the risk of not seeing if
they are central or not if indeed we become stuck on an inconsequential
point.

1) My first concern is the least of them: equivalence of string and
number. Can the natural numbers do something that our string cannot? Is
our string example a suitable example to use in place of natural
numbers? Assuming yes, and that there is no property of natural numbers
that can rescue me from my other concerns - that a string cannot rescue
me from, then I shall continue.

2) I have serious misgivings about the correspondence of a particular
string with a particular sentence. It seems that the only ground for a
correspondence between them is the fact that they occupy the same
space. Otherwise, a string of 50 letters would correspond to any
sentence containing fifty letters, and not just to one particular
sentence. And, we cannot quote particular shared properties between a
string and sentence as a ground for their correspondence because the
statement

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".

would become

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 with the same property as S has the
property "is provable".

What are we to take as the same property? I would not know - it could
be being a theorem, being 'is provable' or being 'a theorom'. It could
be being a string or a sentence.

3) I have another quite serious misgiving. IF a sentence can somehow
be matched or correspond to a string without any problems of the sort
mentioned above, then I need to know either

a) how a sentence can transmutate into a string; AND how the theorem S
can logically address a transmutation if that transmutation is its
foundation, or

b) The nature of that unnamed object that can simultaneously operate
as object, function and theory, or property, string sentence.

As I say, these objections may not be important to the understanding of
G's theorem, and I would not want to pursue red herrings, but I
strongly suspect that they are important to it. On the other hand, I do
not want to miss out further steps.

.

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