On 24 Sep 2015, at 20:49, John Clark wrote:
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,
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. It is not related to the proof of the consistency
of the negation of the axiom of choice made by Cohen, using models of
ZF in which V≠L.
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.
Or set theoretical reality, or analysis, or arithmetic + "arithmetic
is consistent". yes, the arithmetical truth is inexhaustible, ZF knows
much more than PA, and ZF + kappa knows much more than ZF, etc.
> ZF and ZFC see exactly the same arithmetical reality,
In ZFC the Banach-Tarski construction is part of reality, in ZF
it is not.
Yes, it like Euclid's parallel axioms.
If physics is ZFC
ZFC is a set theory. Physics is a theory about a possible physical
reality (primitive or not). You can't equate them.
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.
Performing all computations is already done in the tiny sigma_1 part
of the arithmetical truth, which is so big that no axiomatizable
theory at all can proves its propositions.
>> 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.
I used that as a counter-example.
Bruno
John K Clark
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