Godel, Turing and Truth

John K Clark (johnkc@well.com)
Wed, 15 Jan 1997 11:15:47 -0800 (PST)


On Mon, 13 Jan 1997 "Lee Daniel Crocker" <lcrocker@calweb.com> Wrote:

>Logical systems can have the properties of "consistency" or
>"completeness", but not both.

Yes, so if a system is consistent, that is, if you can't prove that the same
thing is both true and false with it, then it is not complete, and that means
that there are TRUE statements that do NOT have a proof in that system.
That's why I don't understand when you say " Godel does not really have any
implications about the limits of human knowledge".

>"Truth" is something else altogether:


>that's simply how we choose to use the logical system as a
>model of some physical reality.

No. The Goldbach conjecture is either true or it is not, someday a computer
will find an even number that is not the sum of two prime numbers OR it will
not, there is no middle ground. Neither Godel or Turing said otherwise.
The question is, will we ever know if it is true or not? Godel said that, if
not Goldbach then, there are other similar statements that are true but we can
never prove. Turing proved that we can't even identify these hopeless cases,
so we can never tell if the best thing to do is to just give up or whether we
should continue to beat our head against a wall.

Put it this way, if you want the concept of "number" to be meaningful in your
system and you want the ability to count things, then it will include The
Peano Postulates. Is the Goldbach Conjecture consistent with The Peano
Postulates? I don't know, nobody does, and possibly nobody ever will.

John K Clark johnkc@well.com

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