Le 13-juil.-07, à 20:03, Brent Meeker a écrit :

>
> Bruno Marchal wrote:
>>
>> Le 12-juil.-07, à 18:43, Brent Meeker a écrit :
>>
>>> Bruno Marchal wrote:
>>>>
>>>> Le 09-juil.-07, à 17:41, Torgny Tholerus a écrit :
>>> ...
>>>>     Our universe is the result of some set of rules. The interesting
>>>>     thing is to discover the specific rules that span our universe.
>>>>
>>>>
>>>>
>>>>
>>>> Assuming comp, I don't find plausible that "our universe" can be the
>>>> result of some set of rules. Even without comp the "arithmetical
>>>> universe" or arithmetical truth (the "ONE" attached to the little
>>>> Peano
>>>> Arithmetic Lobian machine) cannot be described by finite set of 
>>>> rules.
>>> But it can be "the result of" a finite set of rules. Arithmetic
>>> results from Peano's axioms, but a complete description of arithmetic
>>> is impossible.
>>
>>
>> I don't understand.
>>
>> Let us define ARITHMETIC (big case) by the set of true (first order
>> logical) arithmetical sentences. (like "prime number exist",
>> Let us define arithmetic (lower case) by the set of provable (first
>> order logical) arithmetical sentences, where "provable" means provable
>> by some sound lobian machine.
>> By incompleteness, whatever sound machine you consisder the
>> corresponding "arithmetic" is always a proper subset of ARITHMETIC.
>>
>> So arithmetical truth (alias ARITHMETIC) cannot be described by any
>> finite set of rules. Finite sets or rules can never generate the whole
>> of arithmetical truth.
>>
>> OK?
>>
>> Bruno
>
> Yes, I understand.  But ARITHMETIC is generated by or results from 
> Peano's axioms - right?



Only a tiny part of ARITHMETIC (the set of all true arithmetical 
sentenses, or the set of their godel-number) is generated by the Peano 
Axioms.
Even ZF genererate a little tiny part (but bigger than PA) of 
ARITHMETIC.






>> "existence" is a very very tricky notion. In the theory I am proposing
>> (actually I derived it from the comp principle) the most basic notion
>> of "exists" is remarkably well formalize by first order arithmetical
>> logic, like in Ex(prime(x)):   it exists a prime number.
>
> But isn't this just an elaboration that obscures the prior assumption 
> that numbers exist?


I don't think so. This was clearly assumed at the start. Natural 
numbers are really something you cannot get from less. Actually in 
Peano you can prove the existence of each individual number by proving 
each formula like Ex(x=0), Ex(x = s(0)), Ex(x = s(s(0))) ....


>  If numbers don't exist then Ex(prime(x)) is false, or requires a 
> different interpretation of "E".


Sure. (I am not sure where is your problem)



Bruno






http://iridia.ulb.ac.be/~marchal/


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