# Re: problem of size '10

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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