On 14 Oct 2014, at 04:24, John Clark wrote:
On Mon, Oct 13, 2014 , Bruno Marchal <marc...@ulb.ac.be> wrote:
> Not all information is computable. If (a_i) =
0000001000110000101011000011100010010001101100011101100001101000...
with a_i = 1 if the ith programs (without input) stop, and 0 if not.
That the halting oracle information, and it is not computable.
I know that, I also know that there is no logical reason or
empirical evidence to think that the halting oracle exists in the
physical world or even in Plato's abstract Platonia.
Time implement the halting oracle. There is a result by Schoenfield to
that effect, which shows that the fuincion computable with the halting
oracle are the function computable in the limit (in the computer
scientist sense of limit).
> But, wait, we don't know if there *is* a physical world.
If there isn't a physical world then why do engineers need to spend
millions of man hours and investors spend trillions of dollars
making physical computers? If the existence of digital computers in
Platonia is good enough why bother?
The engineers might be part of the "illusion" too. The question is why
is the "illusion" so stable and so permanent that some can be failed
into believe in a physical ontological universe? And yes, that is part
of the problem, but computer science and mathematics gives clues.
Of course I meant here "primitive physical universe". You don't need
to believe in it to go on the moon.
> if you work in the fundamental, that is a very big assumption,
If you work in the fundamental It's also a big assumption that
Platonia exists.
No, because it is the most least Platonia ever. You need only to
believe that 2+2=4 is true independently of you.
You need already such Platonia to understand "Church thesis",
"computation", etc.
> there isn't even any reason to think it exists in Plato's abstract
Platonia. It's true that Turing prove that there are real numbers,
lots of them, that no computational process can even approximate,
but there is no reason to think anything else can either.
>Wow! Are you suggesting we should abandon the (P v ~P) axioms?
I have no idea what the "(P v ~P) axioms" are and neither I might
add does Google. Does "P" mean Peano, or Polynomial or is it just
more bafflegab? However whatever P means I most certainly don't
reject both P and ~P, because if I were I'd be rejecting everything.
P was a vraiable here. By P v ~P axioms, I mean the scheme of the
excluded middle axioms. (p v ~p, or A v ~A, or x v ~x, with other
notations. It is uses here only for first order closed sentence of
arithmetic, like "2+2=4" or "Ex(x+x=4)", etc.
Bruno
John K Clark
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