🚀 go-pugleaf

In article <350@mtnmath.UUCP> paul@mtnmath.UUCP (Paul Budnik) writes:
>In article <1992Nov13.194334.20447@sun0.urz.uni-heidelberg.de>,
gsmith@kalliope.iwr.uni-heidelberg.de (Gene W. Smith) writes:

>> >This is the question. Continuous models are the simplest to work with
>> >mathematically, but are the simplest possibility as models of physical
>> >reality? I do not think so.

>> It isn't clear to me that any such distinction can be made.

>As one simple example consider the difference between a model based on
>finite difference equations and one based on partial differential
>equations. 

Which is the one you claim is simpler in which department, and why?

>> >There is good reason to suspect that no completed infinite totalities
>> >exist.

>> I know of none.  Name some.

>There are no known physical effects that require completed infinite
>totalities. It is not even clear what such physical effects could be.

So what?  We don't know if the universe is closed or open.  We can't
prove that we really need infinites to do field theory.  None of this
amounts to even a bad reason to suspect infinite totalities don't
exist.

>Mathematicians are as entitled to have speculative philosophical ideas about
>mathematics just as physicists are entitled to have such ideas about physics.
>These ideas are not mathematics they are the philosophy of mathematics.

No doubt.  But the continuum isn't one of them.

>This well known result is called the Lowneheim and Skolem theorem. 

No it isn't.  That is the theorem which tells us that a first order
theory has a countable model.  If you hadn't deleted it, you could
have referred to the bit in my last posting which mentioned it.

Thus, for example, if we write down a first order theory which defines
groups, there must be a countable model for this theory.  This is not
a proof that all groups are countable!  If we write down a first order
theory intended to capture some of the properties of the real numbers,
there will be a countable model for this theory also.  To think that
this shows the real numbers are countable is a mistake so fundamental
it makes me wonder if you have any idea what you are talking about.
Can you explain what it is you are actually trying to say?




-- 
     Gene Ward Smith/Brahms Gang/IWR/Ruprecht-Karls University 
               gsmith@kalliope.iwr.uni-heidelberg.de