random (fwd)

[Lulu of the Lotus-Eaters]

Coming, admittedly, a bit late to the conversation on this topic, I'm
surprised to find Alex a bit too dedicated to his point... and making a
few errors about which he knows better:

Someone wrote:
>A physical RNG can have the property that one cannot make
>any prediction about the value of the next bit even given
>_complete_ information about how the numbers are being
>generated.

"Alex Martelli" <aleaxit at yahoo.com> wrote:
|This says "GIVEN _complete_ information".  Now you say
|you CANNOT "have complete information".  The two
|statements are in contradiction.

No contradiction in this.  "If you COULD have complete information you
can not make a prediction" does not imply that you CAN have complete
information.

If I COULD lift 1000 lbs, I still couldn't lift my car.  But then, I
CAN'T lift 1000 lbs anyway.  Not so complicated.

|I _think_ -- but am not a philosopher or mathematician
|so cannot feel sure -- that part of what Goedel shows is
|that arithmetic is powerful enough to model any other
|formal system.

Well...  I *am* a philosopher, and have occassionally pretended to be a
mathematician (not the most convincing pretense though).  But I don't
think one needs to be to see what's wrong with the above.

Assuming "Goedel" means his incompleteness theorem in the above, he
really doesn't show anything like the above.

The Incompleteness of Arithmetic simply says:

   ANY (finite axiomatic) system powerful enough to model arithmetic
   must contain statements that are not derivable from the axioms of
   that system.

So, for example, ZF can model the Peano axioms, and therefore gets stung
by the incompleteness bug.  But this much leaves open the possibility
that ZF is strictly more powerful than arithmetic.  (question though:
is this actually true of ZF--can it be modelled with the Peano axioms?).

Moreover, it is easy enough to construct a system that is strictly more
powerful than arithmetic, at least if one means something specific and
axiomatic by "arithmetic."

Let's say, for example, that the Peano axioms model arithmetic for our
standard.  Given the Goedel machinery, we can derive a specific statment
of Peano arithmetic, _I_, that is neither a theorem nor a contradiction
of Peano arithmetic.

Therefore, we can create a new system Peano+_I_, that is both consistent
(if arithmetic is) and not modelled by arithmetic.  Moreover, we can now
derive an infinite number of new theorems of Peano+_I_ that utilize _I_
in their derivation and that are independent of Peano arithmetic.

Yours, Lulu...