Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Topic: Formally Unknowability, or absolute Undecidability, of certain arithmetic
formulas.

Replies: 22   Last Post: Jan 29, 2013 8:21 PM

 Messages: [ Previous | Next ]
 namducnguyen Posts: 2,777 Registered: 12/13/04
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:
>
>>>> 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 axiom-system 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 language-structure-centric,
independent of the notion of formal axiom-system.

--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

Date Subject Author
1/27/13 namducnguyen
1/27/13 Frederick Williams
1/27/13 namducnguyen
1/27/13 Frederick Williams
1/27/13 namducnguyen
1/27/13 Jesse F. Hughes
1/27/13 namducnguyen
1/28/13 Jesse F. Hughes
1/28/13 namducnguyen
1/28/13 namducnguyen
1/28/13 Frederick Williams
1/29/13 namducnguyen
1/29/13 fom
1/28/13 Frederick Williams
1/29/13 namducnguyen
1/28/13 ross.finlayson@gmail.com
1/29/13 Michael Stemper
1/29/13 namducnguyen
1/28/13
1/28/13 fom
1/29/13 namducnguyen
1/29/13 fom
1/29/13 Graham Cooper