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