Godel, Turing and Truth

John K Clark (johnkc@well.com)
Sat, 11 Jan 1997 21:12:53 -0800 (PST)


On Fri, 10 Jan 1997 "Lee Daniel Crocker" <lcrocker@calweb.com> Wrote:

>There is nothing that prevents us from forming other axioms
>and other systems of proof that have different unprovables.

That could be very dangerous, it might make your system too powerful,
so powerful it can prove things that are not true. For example, so far
The Goldbach Conjecture has been shown to be true for every number we test it
on, but we can't test it on every number, and so far nobody has found a way
to prove it from the fundamental axioms of number theory. Suppose we give up
trying to prove it correct or having computers look for a counterexample to
prove it wrong, and just add it as an axiom. Consider the possibilities:

1) The Goldbach conjecture is true and it is possible to find a proof from
the current axioms: In this case adding Goldbach as a Axiom is
unnecessary and inelegant.

2) The Goldbach conjecture is untrue but we add it as an axiom: Bad idea,
now you've made mathematics inconsistent. Think of the embarrassment when
some computer finds a number that violates Goldbach, your axiom. Think of
all the mathematics built on top of this dumb axiom that now must go in
the garbage can.

3) The Goldbach conjecture is true but unprovable: In this case it would be
a great idea to add Goldbach as an axiom because then your axiomatic
system would be more complete and just as consistent. The only trouble is,
Turing showed us that if it's unprovable you'll never know it's unprovable,
so it's just too dangerous to take the chance.

On the other hand, Godel doesn't say we can't know anything, he says we can't
know everything. It's the same with Turing's results, sometimes, but not
always, you can prove that a certain result can not be derived from existing
axioms, so then you can add it as an axiom, IF you have courage, IF you think
it is true.

For example, if you take it as an axiom and assume that the parallel
postulate is true, that is, " through any point in the plane, there is ONE ,
and only ONE, line parallel to a given line" then a perfectly consistent
geometry can be made, Euclid did it. Strangely it's also true that If you
assume "Through any point in the plane, there are TWO lines parallel to any
given line" then a perfectly consistent geometry can be made, Lobachevski and
Bolyai did it. It's even true that if you assume " Through a point in the
plane, NO line can be drawn parallel to a any given line" then a perfectly
consistent geometry can be made, Riemann did it. Consistency is not
everything, which one of these geometrys are true must be determined by

The reason all this was possible was because the parallel postulate has been
shown to be independent of the other axioms. An analogous situation has not
been proven for the Goldbach Conjecture. An even number can be found that is
not the sum of two prime numbers OR it can not. There is no middle ground.
I want a system that can prove it is true, but ONLY if it is true.

John K Clark johnkc@well.com

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