On 23 Sep 2015, at 23:59, John Clark wrote:
It seems to me the debate I’v been having with Bruno, the one
about Arithmetic being able to perform calculations all by itself
without the help of matter that obeys the laws of physics, comes
down to the Axiom Of Choice. I would humbly propose that maybe just
maybe mathematics is everything EXCEPT for the Axiom of Choice and
physics is mathematics PLUS the Axiom of Choice If this is true
then for something to be really real and not just sorta real physics
must be able to calculate (choose) it.
The Axiom of Choice says that if you have an infinite number of bins
with two or more different types of things in them then you can
always create a new bin containing exactly one item from each bin.
Bertrand Russell gave this example: “To choose one sock from each
of infinitely many pairs of socks requires the Axiom of Choice, but
for shoes the Axiom is not needed.” With shoes you could have a
finite number of rules (just one in this case) that would work,
always pick the left shoe from each bin, but no corresponding finite
number of rules exists for socks so you’d have to invoke the Axiom
of Choice. This may have some relevance to the following question:
If it exceeds the computational power of the entire universe to
calculate (choose) does the 423rd prime number greater than
10^100^100 really exist or only sorta exist?
To create a bin containing all the integers the Axiom of Choice is
not needed, the 8 Zermelo-Fraenkel Axioms are enough; thus you could
create a bin containing all the integers and only the integers
{1,2,3,4...} , you can also create bins with {2,3,4,5...} and
another with {3,4,5,6...} etc. A finite number of rules (just 8) can
create such bins (sets) . But what about a bin that contains all the
prime numbers and only the prime numbers?
You don't need the axiom of choice to, prove the existence of the set
of prime numbers (and only prime numbers).
Without the Axiom of Choice there is no rule of finite length that
would allow you to choose one and only one prime number from all the
bins I listed above and use them to come up with a new bin
containing all the prime numbers and nothing but the prime numbers.
You can define prime number in arithmetic, and PA can prove their
existence. Everything is even computable. All recursive and
recursively enumerable set are representable already in the theory of
finite sets, or the theory of integers, etc.
Godel proved in 1938 that if you assume the Axiom of Choice is true
then it will cause no contradictions in Zermelo-Fraenkel or in
arithmetic, and Paul Cohen proved in 1963 that if you assume the the
Axiom of Choice is false it will cause no contradictions in Zermelo-
Fraenkel or in arithmetic. In other words the Axiom of Choice is
independent of arithmetic and independent of the Zermelo-Fraenkel
Axioms.
No. Independent of ZF.
That the arithmetical reality is independent of the axiom of choice
has been proved by Gödel using his notion of constructible set. It
shows that ZF and ZFC see exactly the same arithmetical reality, and
so the axiom of choice has no role for proving new arithmetical
theorem. This does not mean that the axiom of choice does not simplify
the search of such truth, but in principle the use of the axiom of
choice can be eliminated. That is a different result than the
independence of AC from ZF (by Gödel and Cohen).
The Axiom of Choice has always been far more controversial than the
8 Zermelo-Fraenkel Axioms, and mathematicians are reluctant to use
it in their proofs unless they have to, in fact it’s almost as
controversial as Euclid’s Fifth Postulate. As I’ve stated it the
Axiom seems intuitively true, almost bland;
It is not an effective axioms. It is a highly non computational axiom,
but then the axiom of infinity also. ZF is doin high level theology
all the times, and has very strong belief.
but the trouble is that you can state the same thing in a different
way that is absolutely equivalent but when stated that way it seems
intuitively false. For example, the Axiom of Choice can also be
stated as "every set can be well ordered” and that seems false;
“well ordered” means it has a least element, it’s easy to see
that the set of positive integers is well ordered but how would you
well order the real numbers? Mathematicians think it’s ugly for the
Axiom Of Choice to produce a set as if by magic with no instructions
on how to actually build it.
OK. And Solovay proved that all set of real numbers is measurable, as
a consequence of the axiom of choice.
We need the choice axiom also to prove the completeness of infinitary
logics (but I avoid them usually).
Another equivalent version of the Choice axiom is: all linear spaces
have a base.
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, but who knows. I would not even been
able to refute this in the computationalist theory, where the axiom of
choice might be handy in the search of the measure on the infinite
computational histories (when seen from the first person perspective
of the machine).
Bruno
John K Clark
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