On 06 Mar 2010, at 23:54, Brent Meeker wrote:
On 3/6/2010 5:41 AM, Bruno Marchal wrote:
On 06 Mar 2010, at 03:02, Brent Meeker wrote:
On 3/5/2010 11:58 AM, Bruno Marchal wrote:
....
In this list I have already well explained the seven step of UDA,
and
one difficulty remains in the step 8, which is the difference
between
a computation and a description of computation. Due to the static
character of Platonia, some believes it is the same thing, but it
is
not, and this is hard to explain. That hardness is reflected in the
AUDA: the 'translation' of UDA in arithmetic. The subtlety is that
again, the existence of a computation is true if and only if the
existence of a description of the computation exist, but that is
true
at the level G*, and not at the G level, so that such an
equivalence
is not directly available, and it does not allow to confuse a
computation (a mathematical relation among numbers), and a
description of a computation (a number).
This mixing of existence and true in the context of a logic confuses
me. I understand you take a Platonic view of arithmetic so that all
propositions of arithmetic are either true or false, even though
most
of them are not provable (from any given finite axioms), so
true=/=provable.
A computation is not true or false. Only a proposition can be true or
false. But the existence of a computation is a proposition.
I was talking about the existence of a computation. This can be
true or
false. Let c be a description of a computation.
The following can be true or false:
"c describes a computation"
That's true ex hypothesi.
OK.
>or "c is a computation"
If I interpret c as a definite description, i.e. name, that's true.
OK.
Otherwise it's false.
Indeed.
or "c is the Gödel
number of a computation"
I suppose that depends on the form used in the description c. If
the Godel numbering scheme is defined and then c is described as
being a certain number in that scheme it's true.
Yes.
Otherwise it's false.
"Ex(x = c & c describes a computation)" == the computation c exists.
OK?
To say that something exists, is the same as saying that an
existential
proposition is true.
But what does it mean to say a computation is true at one level and
not another? Does it mean provable? or it there is some other
meaning
of true relative to a logic?
There is only one meaning of true, in this arithmetical (digital)
frame.
Here by computation I meant a finite computation (to make things
easier). To be a (description of a) finite computation is a decidable
predicate. You can decide in a finite time if c is a computation or
not.
So if a particular computation c exists,
So I should think of c in the above sentence as a description -
distinct from the computation itself. If I informally refer to
computing the largest prime less than 100, is that an example of c
or is it an equivalence class of many different c's.
It is very difficult to explain this, especially by mail. We have the
same difficulty with numbers and actually with all mathematical
notions, and this both when we "explain" a notion to a machine or to a
human.
For example the number 4 is different from "4", "four", "2+2", etc.
But to talk about the number 4 I have to use a description, and then,
automatically, we introduce an ambiguity. With the numbers, we manage
that difficulty very well, by the use and practice. But when we talk
about the conceptual difference between a notion and its
representation, it is harder to explain, given that we have to go from
a description to a description of that description, and hope the
reader will abstract from the first description, which in this
context, necessitates some familiarity or training.
In particular, if I say, let c be computation, I am referring to the
computation itself (an abstract immaterial relation between numbers,
different from any representation of it). It is certainly not an
equivalence class of description. In this case "c" refers to the "real
thing".
PA can prove that fact, and
reciprocally, if PA proves that fact then the computation c exists.
PA, or any sound Löbian machine.
Let us write k for the proposition "c exists". What I just said can
be
written c -> Bc, and Bc -> c. i.e. c <-> Bc.
What happened to k?
I should have written: k -> Bk, and Bk -> k. i.e. k <-> Bk.
Or, I should have directly commit the language trick: let c be the
proposition "c exists".
I recall you that G is the complete logic of provability, PROVABLE by
the machine; and G* is the complete logic of provability, TRUE for
the
machine. As you notice PROVABLE is different from TRUE, and those two
logics are different. Given that we restrict ourself on correct
machine,
we have that G is strictly included in G*.
What I said is that G* proves c <-> Bc (so the existence of a
computation is equivalent with the provability of the existence of a
computation).
But G does not prove c <-> Bc . G does prove c -> Bc (the existence
of a
computation entails the provability of the existence of a
computation),
Certainly for finite computations since you can just perform the
computation to prove it exists.
That is the idea.
but G does not prove Bc -> c. G does not prove that the provability
of
the existence of a computation entails the existence of that
computation.
So in G, (Bc & ~c) does not lead to a contradiction. Can you give a
simple example of such a c in arithmetic?
here is the simplest one:
0 = 1
In arithmetic, B is the Beweisbar predicate of Gödel. By the second
incompleteness theorem B('0=1') is consistent with PA. So "B('0=1') &
~(0 = 1)" does not lead to a contradiction when added as axiom to PA.
You get a theory which is still consistent. You loose soundness, though.
Other examples of such c are:
B('0 = 1')
~ B('0 = 1')
D(0 = 0) (equivalent with ~B(0 = 1))
BB('0 = 1')
B~ B('0 = 1')
G* furnishes an infinity of similar examples, like DBc, DDBc, DDDBc,
etc.
The lesson is: 'existence of a computation' is equivalent with
'existence of a description of a computation', at the G* level. But
not at the G level.
This explains the origin of the abstract symmetry (p -> BDp) and the
quantum behaviors of the "observability logics" (with B = beweisbar(p)
& consistent(0=0)), when p is restricted on the Sigma_1 sentences (the
one whose proofs includes all computations generated by the Universal
Dovetailer).
But this explains also why the 8th step is difficult to understand,
and why the notion of comp-supervenience is a bit tricky. AUDA
explains why a step of UDA is *really* difficult, which explains why I
am so patient with all this. The Löbian machine warned me in advance!
Is it all right?
Bruno
http://iridia.ulb.ac.be/~marchal/
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