Re: Ontology and Physics
Delightful summer everybody,
Quoting "John F. Sowa" <sowa@bestweb.net>:
> Chapter 16 is a good 25-page summary of Cantor's
> set theory, the hierarchy of infinities, Turing
> machines, and Goedel's theorem with good intuitive
> discussion of what they mean.
I have to remind about one central issue concerning the
hierarchy of infinities of ZF, that is basically an
axiomatization Cantor's theory. The continuum hypothesis
argues that the cardinality of the set of natural numbers
is the first infinity, and that the cardinality of the
set of the real numbers is the second infinity, and that
there is no other infinities in between. According to ZF,
the power set of the set of the natural numbers has as
many members as there are real numbers. No one has been
able to prove that continuum hypothesis is right. I don't
know the nature of the attempts, but I have the picture
that either the coherent axioms used in the attempt
could not be used to prove it, or then the axioms that
were used in an othervise succesfull attempt were
uncoherent themselves.
There is a need to prove continuum hypothesis right among
the Cantorists, because if it is not right, then the
hierarchy of infinities is obviously not valid either,
and if it is not good, then ZF has lost quite much of the
ground. That ZF is even partly no-good is something that
most of the mathematicians are not willing to admit.
They take ZF to be valid and that's it. Currently, it is
very normal that math students are making master and
doctor theses about things such as uncountable cardinals,
properties of the the very great cardinals. They must see
something fancy about them. Sure, it sounds great to work
on some super-infinite formations, but this is not a
situation that some have hoped, among then especially
Wittgenstein:
http://uk.geocities.com/frege@btinternet.com/cantor/wittgensteinquotes.htm
> Yet engineers who have a practical problem to solve never
> use anything that remotely resembles a unified theory of
> everything in physics.
I am also very curious about if somebody has used Cantor's
hierarchy to specify something that is actually useful, then
could he or she have specified it also with some other,
more uncontroversial and simpler system(s). If anyone knows
good citations about finitist works, mathematical,
philosophical, or other, please indicate them to me.
If anybody wants to use an uncontroversial set theory without
cardinalities such as infinity+1, and without the empty set,
then I suggest something like the Combined set theory:
http://www.cs.helsinki.fi/u/astyrman/CST.pdf It is now as
simple as it can get, and no one has found any paradoxes in
it. All comments are welcome, and you can try it for free!
A. Styrman