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" or "c is a computation" or "c is the Gödel
number of a computation"
"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, 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.
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), 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.
"c <-> Bc" belongs to the corona G* minus G. It is true, but not
provable by the machine.
OK?
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.
Bruno
http://iridia.ulb.ac.be/~marchal/
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