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
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.
or "c is a computation"
If I interpret c as a definite description, i.e. name, that's true.
Otherwise it's false.
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. Otherwise it's false.
"Ex(x = c & c describes a computation)" == the computation c exists.
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.
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 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
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.
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?
"c <-> Bc" belongs to the corona G* minus G. It is true, but not
provable by the machine.
Once we fix a Löbian machine, we keep the same notion of truth (first
hypostase), and the same arithmetical proposition will be the provable
one. But according to the point of view chosen (the other hypostases),
they obey different logics.
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