On 27 Jan 2014, at 16:12, Brian Tenneson wrote:
Yes, some day a computer might be able to figure out that the set of
rationals is not equipollent to the set of real numbers.
A Lôbian machine like ZF can do that already.
I saw somewhere that using an automated theorem prover, one of
Godel's incompleteness theorems was proved by a computer.
Boyer and Moore, yes, but that is not conceptuallydifferent than ZF,
except that the Boyer-Moore machine uses more efficient sort of AI path.
Gödel discovered that PM already proves his own incompleteness
theorem. All Lôbian machine proves their own Gödel's theorem. They all
prove "If I am consistent, then I can't prove my consistency".
The question I raised initially was this: will there ever be a
machine or human who can correctly answer all questions with a
mathematical theme that have answers?
All? No, for any machine i in the phi_i.
But that is less clear for evolving machines, whose evolution rule is
not part of the program of the machine. Of course, at each moment of
her "life", she will be incomplete, but if her evolution is enough
"non computable", or using some special oracle, it might be that the
machine will generate the infinitely many truth of arithmetic, but not
in any provable way.
I didn't think so in my original post but now I'm starting to
wonder. It's the existence of undecidable statements that would
probably lead to the machine or human not being able to do it in
general. This reminds me of the halting problem.
Those are related. Undecidable is always relative. Consistent(PA) is
not provable by PA, but is provable in two lines in the theory PA
+con(PA). Of course PA+con(PA) cannot prove con(PA+con(PA)).
What about PA+con(I), with I = PA+con(I). It exists as we can
eliminate the occurence of I by using the Dx = "xx" method. Well, in
this case PA+con(I) can prove its own consistency, but only because it
is actually inconsistent.
The good news is we will never run out of mathematical territory to
think about.
Yes indeed, even if we confine ourselves on elementary (first order)
arithmetic. There is an infinity of surprises there.
Bruno
On Mon, Jan 27, 2014 at 6:58 AM, Gabriel Bodeen
<[email protected]> wrote:
FWIW, under the usual definitions, the rationals are enumerable and
so are a smaller set than the reals. I'd suppose that if people can
figure that out with our nifty fleshy brains, then a well-designed
computer brain could, too.
-Gabe
On Friday, January 24, 2014 1:23:40 AM UTC-6, Brian Tenneson wrote:
There are undecidable statements (about arithmetic)... There are
true statements lacking proof. There are also false statements about
arithmetic the proof of whose falsehood is impossible; 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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