On Thu, Sep 24, 2015 at Bruno Marchal <[email protected]> wrote: > > You can define prime number in arithmetic, >
Who cares? I'm not interested in arithmetic or in anything else defining prime numbers, I'm interested in *CALCULATING *prime numbers. > > That the arithmetical reality is independent of the axiom of choice has > been proved by Gödel > Paul Cohen not Godel proved that arithmetical reality is independent of the Axiom of Choice, Godel just proved it was consistent with it, Cohen proved its negation was consistent with it too. And if arithmetical reality is independent of the axiom of choice then something that was dependent on BOTH arithmetical reality AND the Axiom of Choice would be different from just arithmetical reality, maybe something like physical reality. > > > ZF and ZFC see exactly the same arithmetical reality, > In ZFC the Banach-Tarski construction is part of reality, in ZF it is not. If physics is ZFC then Banach-Tarski is physical reality even if it's not arithmetical reality and physics can do stuff that arithmetic can't; but we already knew that, arithmetical reality isn't sufficient to perform calculations. >> > Also if the the Axiom Of Choice is true then the Banach-Tarski > construction (sometimes called paradox) can be done. If you cut up a solid > sphere and then put all the pieces back together in a way specified by > Banach and Tarski you can create TWO solid spheres of a size equal to the > original single sphere. This can’t happen in the real physical world so > does this fact work against my idea that Physics is arithmetic plus > the Axiom Of Choice? Maybe not because maybe it does happen in the real > physical world. We know from astronomical observation that space is > expanding, new space is being created, and maybe Banach-Tarski is how > physics does it. > > That seems quite speculative > It is, but not as speculative as the idea that the human biological brain needs dark matter to operate. 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.
