Eliah Kagan wrote:
Duncan Patton a Campbell wrote:
When you mathematically formalize a physical theory, you use a system
that is powerful enough to formulate the theory's physical claims.
That system is also going to be powerful enough to formulate all sorts
of other stuff, things that have nothing to do with physics. Suppose
one extends Zermelo-Frankel Set Theory (or designs a system that works
like it) to do physics. I think the undecidability of the Axiom of
Choice would be a poor example in this specific case because I believe
the Axiom of Choice is used in topology, and therefore may be valuable
in doing physics.
Be careful. Axiom of choice is used all over but theoretically one
could pin point results that depend on it.
Very careful mathematicians will do. Some results like existence of
non-measurable sets in classical Lebesgue measure theory
can not be proved without Axiom of choice. (that fact is the very hard
theorem in its own right)
But the Continuum Hypothesis--the claim that there
are no sets cardinally bigger than the set of integers but cardinally
smaller than the set of reals-
That is NOT true. Late Poul Cohen got a Filds medal for proving
something like this (I will get myself now in to BIG trouble because
of imprecise statements I am making to describe the results)
Roughly, the continuum hypothesis in mathematics has the same position
as the fifth euclidean postulate i.e.
You could build consistent mathematics if you assume that there are no
sets of bigger cardinality than alef0 and less than c.
(Like assuming Euclidean axiom of parallelility you get Euclidean
geometry.)
You could also build consistent mathematics assuming that there are sets
with cardinality bigger than alef0 and less than c roughly corresponding
to the case of geometry of Lobachevsky-Bolyai when an alternative axiom
of parallelility is added on the top of
axioms of incidence, congruence, order, and continuity to build geometry
of Lobachevsky.
The further discussion is definitely out-side of the scope of this
mailing list.
-probably has nothing to do with
physics. So your physical formalization can be used to formulate the
Continuum Hypothesis, which is undecidable, but since that's not
actually *about* physics, it doesn't make your system incomplete as a
formalization of physics. Now, often ideas in abstract mathematics
turn out to be useful in applied fields, and I would not discount the
possibility that transfinite arithmetic might turn out to be
applicable in physics, though I can conceive of no way that it would.
But so long as there is *some* formulable question arising out of the
math you use to do your physics that is itself not physically
important, you can have a formal system that, as a formulation of
physics, is complete and consistent.
-Eliah
I also notice very uncareful use of the Russell paradox on this mailing
list.
Godel's results are invoked very uncarefully as I noticed earlier and
as the content of the above mail points out by
giving little bit more details about formal systems used in mathematics.
Kind Regards to Everyone
Predrag
