Date: Jan 29, 2013 2:42 AM
Author: Graham Cooper
Subject: Re: Formally Unknowability, or absolute Undecidability, of certain<br> arithmetic formulas.
On Jan 28, 3:22 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> In some past threads we've talked about the formula cGC
> which would stand for:
> "There are infinitely many counter examples of the Goldbach Conjecture".
> Whether or not one can really prove it, the formula has been at least
> intuitively associated with a mathematical unknowability: it's
> impossible to know its truth value (and that of its negation ~cGC) in
> the natural numbers.
> The difficulty to prove such unknowability, impossibility, is that
> there are statements that are similar in formulation but yet are
> known to be true or false. An example of such is:
> "There are infinitely many (even) numbers that are NOT counter
> examples of the Goldbach Conjecture".
> The difficulty lies in the fact that there have been no formal
> logical way to differentiate the 2 kinds of statements, viz-a-viz,
> the unknowability, impossibility.
> In this thread, we propose a solution to this differentiation
> difficulty: semantic _re-interpretation_ of _logical symbols_ .
> For example, we could re-interpret the symbol 'Ax' as the
> Specifier (as opposed to Quantifier) "This x", and 'Ex' as
> the Specifier "That x". And if, for a formula F written in L(PA)
> (or the language of arithmetic), there can be 2 different
> "structures" under the re-interpretations in one of which F is true
> and the other F is false, then we could say we can prove
> the impossibility of the truth value of F as an arithmetic
> formula in the canonical interpretation of the logical
> symbols 'Ax' and 'Ex'.
> (Obviously under this re-interpretation what we'd mean as a language
> "structure" would be different than a canonical "structure").
> Again, this is just a proposed solution, and "This x" or "That x"
> would be not the only choice of semantic re-interpretation.
> As long as the semantic re-interpretation makes sense, logically
> at least, it could be used in the solution.
> But any constructive dialog on the matter would be welcomed and
> appreciated, it goes without saying.
> There is no remainder in the mathematics of infinity.
> NYOGEN SENZAKI
the only UN-PROVABLE sentences are idiotic nonsense like
X = 'not (exist( proof( X )))'
Really UN-PROVABLE, UN-COUNTABLE, UN-COMPUTABLE
are ALL Superfluous Self Inflicted Diatribe!
The only WITNESS to missing computable reals is CHAITANS OMEGA!
based on :
S: if Halts(S) Gosub S
Just use the HALT values to make a POWERSET(N) instead!
x e P(N)_1 IFF TM_1(x) Halts
Now it proves a powerset N *IS* countable!
Really, abstract mathematics is the biggest century long con to ever
exist under the guise of 'WEVE FORMALLY PROVED IT ALL!'
You haven't formally proven ANY OF ALL THE UN-DOABLE RUBBISH!
You redid the same errors with Calculus and BIJECTION / ONTO self
defeating function definitions instead!
|N| = |GODEL NUMBERS| = |FUNCTIONS|
= |CHOICE FUNCTIONS| = |SETS|
by your own AOC.
You don't have a SINGLE INFINITE LENGTH FORMULA to even have un-
countable many functions - the whole notion of un-representable
functions is an oxy moron.