Le 05-déc.-08, à 20:51, Abram Demski a écrit :
>
> Bruno,
>
> Are you asserting this based on published findings concerning
> provability logic? If so, I would be very interested in references. If
> not, then your results obviously seem publishable :).
I have published this in french a long time ago, but then I have
discovered that it has been publishe before by Montague and Kaplan (see
also Thomason). It is related to the fact that Knowledge, like truth
(cf Tarski), is not definable through an arithmetical predicate. In
"conscience and mécanisme" I illustrate a similar fact by using
(informally) the Lowenheim Skolem theorems.
Then I think the provability logic put a immense light on this, in a
transparently clear (arithmetical) frame, and that is a big part of my
thesis (the AUDA part).
> That is, if you
> can show that huge amounts of set theory beyond ZFC emerge from
> provability logic in some way...
I guess I have been unclear, because I am not saying that. I am saying
the more obvious (once we are familiar with incompleteness,
indefinissability, uncomputability etc) fact that a machine can infer
true but unprovable (by her) things about herself. It is just that a
provability machine, having furthermore inductive inference abilities
will generate more truth about itself than those which are provable by
the machine.
>
> Anyway, I'd definitely be interested in hearing those ideas.
Those ideas constitute the AUDA part. It is an abstract translation of
UDA in the language of "the" universal machine. It is needed to extract
constructively physics from computer science. I only get the
propositional physics (which is a billionth of "real" physics, yet I
got both the communicable physical logic and the uncommunicable
physical logic, that is both the quanta and the qualia. In that sense
it is already more than "usual physics", which (methodologically or
not) put the qualia and its subject under the rug.
Bruno
>
> --Abram
>
> On Fri, Dec 5, 2008 at 4:20 AM, Bruno Marchal <marchal.domain.name.hidden>
> wrote:
>>
>>
>> On 05 Dec 2008, at 03:56, Russell Standish wrote:
>>
>>>
>>> On Wed, Dec 03, 2008 at 04:53:11PM +0100, Bruno Marchal wrote:
>>>>
>>>> I really don't know. I expect that the mathematical structure, as
>>>> seen
>>>> from inside, is so big that Platonia cannot have it neither as
>>>> element
>>>> nor as subpart. (Ah, well, I am aware that this is
>>>> counter-intuitive,
>>>> but here mathematical logic can help to see the consistency, and the
>>>> quasi necessity with formal version of comp).
>>>>
>>>
>>> This point rather depends on what Platonia contains. If it contains
>>> all sets of cardinality 2^{\aleph_0}, then the inside view of the
>>> deployment will be conatained in it.
>>
>> I am not sure. In my opinion, to have a platonia capable of describing
>> the first person views emerging from the UD entire work, even the
>> whole of Cantor Paradise will be too little. Even big cardinals (far
>> bigger than 2^(aleph_0)) will be like too constrained shoes. Actually
>> I believe that the first person views raised through the deployment
>> just escape the whole of human conceivable mathematics. It is big. But
>> it is also structured. It could even be structured as a person. I
>> don't know.
>>
>>
>>>
>>>
>>> I do understand that your concept of Platonia (Arithmetic Realism I
>>> believe you call it) is a Kronecker-like "God made the integers, all
>>> the rest was made by man", and so what you say would be true of that.
>>
>>
>> Yes the 3-Platonia can be very little, once we assume comp. But the
>> first view inside could be so big that eventually all notion of 1-
>> Platonia will happen to be inconsistent. It is for sure unameable (in
>> the best case). I discussed this a long time ago with George Levy: the
>> first person plenitude is big, very big, incredibly big. Nothing can
>> expressed or give an idea of that bigness.
>>
>> At some point I will explain that the "divine intellect" of a lobian
>> machine as simple as Peano-Arithmetic is really far bigger than the
>> "God" of Peano-Arithmetic. I know it is bizarre (and a bit too
>> technical for being addressed right now I guess).
>>
>> Have a good day,
>>
>> Bruno
>>
>>
>> http://iridia.ulb.ac.be/~marchal/
>>
>>
>>
>>
>>>
>>
>
> >
>
http://iridia.ulb.ac.be/~marchal/
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Received on Sat Dec 06 2008 - 09:08:03 PST