On 30 May 2012, at 18:31, John Clark wrote:
On Wed, May 30, 2012 Bruno Marchal <[email protected]> wrote:
> The axiom of choice is not a physical law.
That is true, but it is consistent with empirical physical evidence
about how the universe works. In non-mathematical language the Axiom
of Choice says that every event need not have an associated cause,
and like all good axioms it is not intuitively obvious that its
opposite must be true. And as a bonus it appears that our Universe
follows the Axiom of Choice, some things really do happen for no
reason, some things are random.
The axiom of choice just asserts that an arbitrary product of a family
of non empty set is non empty. It is equivalent with the statement
that all set can be well ordered. It is trivial on finites sets, and
looks rather intuitive on infinite sets.
There is no clue of direct relationship with physics, other than
assuring (another consequence) that infinite linear spaces have alway
a base, and things of that type.
It has a priori nothing to do with free will, neither with physics,
and the term "choice" has no serious relation with the notion of human
choice.
Interestingly ZF and ZFC which proves much more arithmetical sentences
than PA, are actually equivalent, i.e. proves the same arithmetical
sentences. The choice (C) add nothing to arithmetic from ZF pov. This
is not obvious and the proof relies on Gödel constructible universes.
I refer you to a good book on axiomatic set theory.
This does not mean I am not using it, as you need it to prove the
completeness of the predicate logic, which uses the fact that each
consistent set of first order sentences can be extended into one
maximal consistent set. This needs the axiom of choice. This allows
the fundamental relation (consistent <=> having a model), and (proof
<=> true in all models).
Bruno
John K Clark
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