On 03 Feb 2014, at 12:09, Kim Jones wrote:
On 3 Feb 2014, at 7:00 pm, Bruno Marchal <marc...@ulb.ac.be> wrote:
I can imagine a semi-block universe in which, as you've often
remarked, the past is a block and the universe keeps adding new
moments and growing. This would be like Barbour's time capsules,
except just sticking everything into one capsule, like a history
book that keeps adding pages. But yes it implies another exterior
"time" in which this "happens"; but then so does Bruno's UD.
Only if you call the order of the natiral numlbers a "time". The UD
does not use anything more.
The UD both generates and executes all programs.
OK.
In other words it reads the numbers, yes?
I am not sure what you mean.
It both reads the numbers to generate numbers
In the language Prolog, you can easily write a program which generates
the numbers, and accessorily is also a statement in predicate logic:
(:= is the same as "<-", or "only if")
Number(0)
Number(S(x)) := Number(x)
You just type this on your computer, when it executes a Prolog
interpreter, and then you type
Number(x)? print(x)
Then the prolog interpreter generates, without stopping
0
S(0)
S(S(0))
S(S(S(0)))
S(S(S(S(0))))
S(S(S(S(S(0)))))
S(S(S(S(S(S(0))))))
S(S(S(S(S(S(S(0)))))))
S(S(S(S(S(S(S(S(0))))))))
S(S(S(S(S(S(S(S(S(0)))))))))
S(S(S(S(S(S(S(S(S(S(0))))))))))
S(S(S(S(S(S(S(S(S(S(S(0)))))))))))
S(S(S(S(S(S(S(S(S(S(S(S(0))))))))))))
...
By typing CTRL-Q, fortunately, you can stop the prolog interpreter, so
that you can do something else.
We could also not have written "print(x)". In that case the prolog
interpreter would still generate the numbers, but without printing it
on the screen. Such a program does not compute a function, it
generates a set. Thinking in term of the phi_i, I would say that such
a program has no output, even when it prints his finding of objects
satisfying the axioms given.
I am not sure he read numbers. It is asked if it exists x such that
number(x), by the "?", and as he believes number(0), he find quickly
an example. Then the second axiom is verified for x = 0:
"number(S(0)) := number(0). And so by modus ponens, knowing number(0),
it concludes that number(S(0)), and find another solution S(0), and he
get the next numbers by applying, again and again, the modus ponens
rule.
and reads them to execute them. I mean the order of the natural
numbers is itself a number isn't it?
Somehow Plotinus asked the same question!
Well, in this case, the order is not a number per se. The usual
mathematical content of the order "<" of the natural number s is the
infinite set {(0, 1), (0, 2), (1, 2), (0, 3), (1, 2), (1, 3), (0,
4), ... } (the set of all (x, y) such that x < y. It is an infinite
set. But it is computable, so there is a number o "order" so that
phi_o(x,y) = 1 if x < y and 0 else. The order (the infinite set) is a
number, like you are a machine. It is only in virtue of being
represented by a universal machine/numbers.
Then there are just numbers READING ie glimpsing, looking at,
noticing or whatever - each other. What has time got to do with any
of that?
OK. So a universal dovetailer does not just generate the numbers, it
executes them on each of them.
It generates all i, i = 0, 1, 2, 3, ... (or 0, s(0), etc... depending
of its language), but it generates also all the P_i, that is the
programs, ordered lexicographically, in some language (may be its own
language, that is not important), and it computes all phi_i, that is
generates the one step dynamical computation of P_i(j), for all i and j.
Now, arithmetic itself is Turing universal, so that all this is done
in he "block-possible-activities-of-the-universal-numbers" which is
basically the sigma_1 complete part of arithmetical truth.
Computation is a dynamical notion, but its dynamic is digital and
entirely defined by some natural number order.
You can always refer to the nth step of the computation of this or
that machine, done by the UD, or to the nth step of the UD itself. It
just happen that such a dynamical notion, a bit like F= Ma, has a
complete "block-universe" depiction in a tiny and effective (sem-
decidable, partial computable) part of the atemporal arithmetical
reality, the realm which satisfies propositions like 2+2=4 or Fermat
theorem.
What did you mean by "reading numbers"?
Bruno
Kim
============================
Kim Jones B.Mus.GDTL
Email: kimjo...@ozemail.com.au
Mobile: 0450 963 719
Landline: 02 9389 4239
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"Never let your schooling get in the way of your education" - Mark
Twain
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