On Fri, Sep 25, 2015 at 11:36 AM, 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, and that proves the > consistency of AC. > The consistency of AC does not enter into it, the independence of AC from ZF does. 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. >> >> >> If physics is ZFC > > > > > ZFC is a set theory. > I know. > > > 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. > > Performing all computations is already done in the tiny sigma_1 part of > the arithmetical truth, > 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. 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. I'm not sure exactly what it's lacking, maybe it's the Axiom of Choice and maybe it's something else, but it sure as hell is lacking SOMETHING because nobody has been able to start a computer hardware company with zero manufacturing costs. 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.
