On 27 Jan 2014, at 19:55, John Clark wrote:
On Fri, Jan 24, 2014 at 2:23 AM, Brian Tenneson <[email protected]>
wrote:
> There are undecidable statements (about arithmetic)... There are
true statements lacking proof.
Yes.
> There are also false statements about arithmetic the proof of
whose falsehood is impossible;
A proof is a FINITE number of statements establishing the truth or
falsehood of something;
Not establishing the truth, but establishing the theoremhood.
At the metalevel, if the theiry is formalized in first order logic,
you will have that the proved proposition will be satisfied in all
models of the theory, but this cannot, in general, be shown *in* the
theory, unless the theory is Löbian (but that is not easy to prove---I
don't use that).
if Goldbach's Conjecture is untrue then there is a FINITE even
number that is NOT the sum of 2 primes.
Yes. The negation of Goldbach is Sigma_1. Goldbach is Pi_1. Like
Riemann Hypothesis.
But Syracuse conjecture is above Pi_1. You cannot decide it, nor his
negation, by a simple mechanical procedure.
It would only take a finite number of lines to list all the prime
numbers smaller than that even number and show that no two of them
equal that even number, and that would be a proof that Goldbach's
Conjecture is wrong.
The real problem would come if Goldbach's Conjecture is true (so
we'll never find two primes to show it's wrong) but can not be
proven to be true (so we will never find a finite proof to show its
correct).
OK.
Bruno
John K Clark
not just impossible for you and me but for a computer of any
capacity or other forms of rational processing. We'll never have a
computer, then, that will work as a mathematically-omniscient
device. By that I mean a computer such that every question that has
a mathematically-oriented theme having an answer truthfully can be
answered by such a device. Calculators demonstrate the concept but
are clearly not mathematically-omniscient: you ask the calculator
what is 2+2 and press a button and "presto" you get an answer. What
I'm talking about would be questions like "is the set of rational
numbers equal in size to the set of real numbers", and get the
correct answer. So we will never have such a computer no matter what
its capacities are, even if computer encompasses the entire human
brain. Unfortunately, that means that even for humans, we will never
know everything about math. Unless something weird would happen and
we suddenly had infinite capacities; that might change the
conclusions.
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