Le 12-juil.-07, à 18:43, Brent Meeker a écrit :

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>
> 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
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