On Sat, Sep 26, 2015 Bruno Marchal <[email protected]> wrote: > >>> >> >>> Paul Cohen not Godel proved that arithmetical reality is independent of >>> the >>> >>> Axiom of Choice >> >> >> > > >> I don't think so. The independence of arithmetic from AC in ZF follows >> from Gödel's proof that V=L -> AC. A model of ZF where all sets are >> "constructible" (V = L) verifies the choice axiom >> > If a axiom has been verified, that is to say if it can be derived from other axioms, then it no longer needs to be a axiom and is just the result of more fundamental axioms. Paul Cohen proved that AC can not be derived from ZF.
>> >> Godel proved that if you assume that AC is true ZF will produce no >> contradictions, 25 years later Paul Cohen >> proved that if you assume AC is false ZF will STILL not produce >> any contradictions, and so AC must be independent of ZF and can not be >> derived from ZF. >> > > > > Yes, but this has nothing to do with what I am saying. The fact that the > arithmetical truth is independent of the choice axiom > Then a lot of stuff that mathematicians think is true is not true, or at least can't be proven to be part of " arithmetical truth " because the Axiom of Choice is needed to prove them. > > > can be seen as a corollary of Gödel's proof that AC is consistent with ZF, > Godel proved in 1938 that AC was consistent with ZF but for all Godel knew The Axiom of Choice could be derived from Zermelo-Fraenkel; and that is in fact what Godel believed at the time and what most mathematicians thought, even Paul Cohen thought so and was as surprised as anyone when he found in 1963 that the negation of AC was consistent with ZF too and thus independent of ZF. > > >>> >> >>> Physics is a theory about a possible physical reality >> >> > >> >> >> I know. So if physical reality >> is ZFC ( a big "if" I admit but it could be) then physical reality has >> something that arithmetic derived from just ZF does not have. >> > >> > "physical reality is ZFC" means nothing to me. > Physical reality is Zermelo-Fraenkel plus the Axiom of Choice, " arithmetical truth " is just Zermelo-Fraenkel . I'm not saying it's true, I'm just saying that's what it means; it might be wrong but it's not gibberish. > > Anyway, I do not assume that there is a physical reality. > Hmm. Margaret Fuller once said "I accept the universe" to which Thomas Carlyle replied "Gad, you 'd better". Unlike you at leas Fuller accepted the universe, I wonder what Carlyle would say to you. > >> >> And yet despite repeated requests you are unable or unwilling to explain >> why you can't start the >> >> Tiny >> >> Sigma_1 Computer Hardware Corporation and become the richest man on the >> planet. > > > > > N > > ot at all. I think you don't read the answer. The answer, I repeat again, > is that to build an hardware corporation I need hardware > Yes perfectly true, you need physical hardware. But my question is* WHY*? The only answer can be that physical hardware has something that "arithmetical truth" does not. We may not be certain what that something is but the fact that computer hardware companies have non zero manufacturing costs is proof that one has something the other does not. > > > and I need to implement the universal machine in that hardware. > Yes exactly you need to implement it, but to implement it mathematics needs help, it needs physics! > > > to have a computation running, I need only arithmetic. > But it is a fact that to have a successful company that provides answers to arithmetical problems arithmetic is *NOT* all you need. > > Numbers ==> computations ==> dreams ===> physical reality ===> physical > computation ===> hardware company > OK, but the hardware company certainly has access to numbers so why doesn't INTEL just make calculations directly and forget about all that unnecessary and expensive messing around with silicon? > >> >> You can't explain it but I can, you can't do it because a physical >> silicone microprocessor chip has something that Robinson arithmetic >> and >> "the tiny sigma_1 part of the arithmetical truth" >> >> lacks. > > > > Yes, that is right. But that things which is lacking is an illusion, > An Illusion is a perfectly respectable subjective phenomenon, and so is consciousness; so you're saying that subjective phenomenon is the thing that that matter that obeys the laws of physics can create that arithmetical truth can not create. Well maybe, arithmetical truth is certainly lacking something that physics has. > > > like someone can emulate Einstein's brain > Then that emulation is Einstein. > > > making a course in GR without any understanding of GR. > Then Einstein didn't "understand" GR and like "God" and more recently "theology" the word has lost all meaning. This destruction of words you're engages in is getting scary, pretty soon we'll just have grunts. >> >> I'm not sure exactly what it's lacking, maybe it's the Axiom of Choice >> and maybe it's something else, > > > > It is the primitive matter which is lacking. > Then primitive matter is more fundamental than arithmetic. QED. >> >> but it sure as hell is lacking SOMETHING because nobody has been able to >> start a computer hardware company with zero manufacturing costs. > > > > > But the hardware and the primitive matter are explained in RA, > Can RA also give an answer that INTEL stockholders would accept to explain why shutting down all their silicon chip fabrication plants and just ordering their employees to meditate about numbers didn't turn out to be a wise business move? John K Clark > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
