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

##
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Yes, I understand. But ARITHMETIC is generated by or results from Peano's
axioms - right?
Brent Meeker
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