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        riddles >> hard >> Peano, Godel and Goldbach
        
(Message started by: Mickey1 on Dec 28^th, 2011, 7:40am)

Title: Peano, Godel and Goldbach
Post by Mickey1 on Dec 28^th, 2011, 7:40am

It is sometimes speculated that Goldbach’s conjecture could be a true but improvable statement, such as mentioned in Godel’s theorem of incompleteness.
This is an attempt to formulate a counterargument:

1. Consider two series of natural number, i.e. both from 1 to infinity.
2. Around the first series we build an axiom system such as Peano’s, called unrestricted.
3. Around the second, we consider only the axioms related to addition, not multiplication. It is therefore called restricted (avoiding the words completed and uncompleted because of these words’ connotation to Godel’s proof).
4. We now make a one-to-one correspondence between the two series such that the number n in then unrestricted series corresponds to the number n in the restricted series.
5. We then use this correspondence to denote some elements in the restricted series “prime numbers”, keeping in mind that in the restricted series, prime numbers have no real meaning.
6. We now start a project as a term of reference, to test whether the different formulations of Goldbach’s conjecture(s) are correct. This can be done since the conjectures are limited to addition processes.
7. We note from Wikipedia that “For each statement in Presburger arithmetic, either it is possible to deduce it from the axioms or it is possible to deduce its negation” (Presburger Arithmetic).
8. The components used in proofs within the restricted system would be a valid subsystem of the corresponding elements of the unrestricted system.
9. It seems therefore that Goldberg’s conjecture(s) cannot be of a Godel undecidable type(s).

Title: Re: Peano, Godel and Goldbach
Post by towr on Dec 28^th, 2011, 11:17am

Since you don't define the prime numbers without the help of an arithmetically complete system, the argument breaks down halfway through. You would need to be able to define prime numbers within the restricted system itself for it to work, not import them from the unrestricted one. Because in the latter case you're simply working in the larger, combined system where you have both.

Title: Re: Peano, Godel and Goldbach
Post by Mickey1 on Dec 29^th, 2011, 2:01am

I guess you are right. I had a bad feeling about importing the primes.