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?
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.
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.
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; 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.
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.
John K Clark
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