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Re: Formally Unknowability, or absolute Undecidability, of certain arithmeticformulas.
Posted:
Jan 29, 2013 8:21 PM


On 29/01/2013 11:29 AM, Michael Stemper wrote: > In article <xbfNs.425$OE1.376@newsfe26.iad>, Nam Nguyen <namducnguyen@shaw.ca> writes: >> On 27/01/2013 12:07 PM, Frederick Williams wrote: >>> Nam Nguyen 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. >>> >>> No one thinks that but you. >> >> If I were you I wouldn't say that. Rupert for instance might not >> dismiss the idea out right, iirc. >> >>> Its truth value might be discovered tomorrow. >> >> You misunderstand the issue there: unknowability and impossibility >> to know does _NOT_ at all mean "might be discovered tomorrow". > > Are you implying that GC have been proven to be indepedent of the usual > axioms of number theory?
No. We don't even know if any usual axiomsystem for the natural numbers (e.g. PA) is syntactically consistent, or inconsistent (in which all formulas would be provable).
For the record, I've always maintained that the issue of impossibility to know of the _truth value_ of cGC is languagestructurecentric, independent of the notion of formal axiomsystem.
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI 



