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Re: Formally Unknowability, or absolute Undecidability, of certainarithmeticformulas.
Posted:
Jan 28, 2013 10:04 PM


On Jan 27, 3:41 pm, Nam Nguyen <namducngu...@shaw.ca> wrote: > On 27/01/2013 1:02 PM, Frederick Williams wrote: > > > > > > > > > > > Nam Nguyen wrote: > > >> 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". > > >> It's impossible to know of a solution of n*n = 2 in the naturals > >> means it's impossible to know of a solution of n*n = 2 in the naturals. > >> Period. > > >> It doesn't mean a solution of n*n = 2 in the naturals "might be > >> discovered tomorrow", as you seem to have believed for a long time, > >> in your way of understanding what unknowability or impossibility > >> to know would _technically mean_ . > > > I am not talking about the words 'unknowability' and 'impossibility to > > know' the meanings of which I know. Nor am I talking about 'It's > > impossible to know of a solution of n*n = 2 in the naturals.' I'm > > talking about 'There are infinitely many counter examples of the > > Goldbach Conjecture'. > > Ok. So you seem to be saying that (unlike the lone Nam Nguyen) everyone > should not think that it's impossible to know the truth value of cGC > since "its truth value might be discovered tomorrow", according to your > knowledge about mathematical logic. > > But, A) what's the technical definition of "might be discovered > tomorrow"? "Tomorrow" relative to which side of the International > Date line? The Australia side? or the US side? And B) what happens > if before "tomorrow" has arrived, "today" somebody would discover > the truth value of cGC, rendering "might be discovered tomorrow" > _meaningless_ ? > > I meant, what would "tomorrow", "today" have anything to to with > _mathematical logic_ ? And, would you have a concrete proof that its > truth value "might be discovered tomorrow"? > > How do you know that it's _not_ impossible to know the truth value > of cGC? > > All that aside, this thread ultimately is about an example of > noncanonical interpretation of the semantic of logical symbols > in general. > > Would you be in the position to offer some evaluation, insight, on > such noncanonical interpretation? > >  >  > There is no remainder in the mathematics of infinity. > > NYOGEN SENZAKI > 
If it were proven independent of a given set of axioms or rules about inference in number theory, then it wouldn't have a truth value because there are extensions where it is true and extensions where it is false, where it would. Here that statement reflects basically a liberal consideration of even what is "true" about the natural integers: for example in their infinitude whether there are simply infinite members, or as better known in number theory a compactifying point at infinity, or in the general naive view that there isn't, or in the even more naive that there aren't.
Then, there either are or aren't counterexamples or infinitely many counterexamples to a Goldbach conjecture, given particular rules or axioms of what the numbers, as objects, under consideration, are. Where cGC _is_ yet a conjecture, that is the plain statement that its truth value is among true, false, and independent of the system under consideration, it _is_ one of those.
Found this of interest:
http://www.logic.amu.edu.pl/images/9/95/Pogonowski10vi2010.pdf
Basically any incomplete theory would never be categorical: a theory of everything would be complete, it won't be a regular set theory. A similar notion is that only true statements are in the language of the theory, Epimenides' paradox is instead Janus' introspection.
Regards,
Ross Finlayson



