Pat, (01)
Although I do not share Avril's desire to outlaw transfinite
theories, I also believe that there is no reason to require
that theories of foundations of classical mathematics must
assume transfinite entities. (02)
> ... it is hard to see how one can rationally deny that it is
> possible to make indefinitely fine discriminations of location
> on the continuum. (03)
I recommend Tarski's axioms for solid geometry, in which the only
primitives are spheres of arbitrary, but always finite radius.
It's a short and eminently readable paper: (04)
Tarski, Alfred (1929) "Foundations of the geometry of solids,"
in Tarski (1982) pp. 24-29. (05)
Tarski, Alfred (1982) _Logic, Semantics, Metamathematics_,
Second edition, Hackett Publishing Co., Indianapolis. (06)
After presenting his axioms and developing some of their
implications, Tarski defined the word 'point' as the limit
of an infinite sequence of nested spheres of decreasing
radius. He then showed how Euclid's axioms stated in terms
of points could be modeled as such infinite sequences of
spheres. As a result, all of Euclidean geometry could be
built up in terms of a foundation consisting of spheres. (07)
There are many ways of interpreting this exercise. My preference
is to view the spheres as a better model of physical reality
than the points. It is fairly easy to generalize Tarski's
spheres to arbitrary solid shapes that approximate physical
atoms and molecules. That view could be called "reality". (08)
The Euclidean language could be viewed as an idealization or
abstraction away from "reality" into a theoretical view that
might have some computational advantages, but would not be
considered an accurate model of "the way things are". (09)
> So if a continuum exists at all, then a point continuum exists
> alongside it, as it were. (010)
One could say that. But I would prefer to say that Tarski's
spheres are a more faithful physical model, and the language
about points is an approximation that might be useful for
computational purposes. But they would be two very distinct
languages. In Tarski's basic axioms, the parts of spheres are
other spheres. Points belong to a very different language,
and it would be a mistake to say that points are "parts"
of spheres. (011)
In that sense, spheres are the physical primitives, and points
are the *limits* of descending sequences of spheres. Spheres
are ontologically prior, and points are theoretical abstractions. (012)
> What distinguishes a potential infinity from infinity? (013)
I would not recommend that way of talking. Another way to state
the matter is to say that spheres are the ultimate geometrical
primitives. Any physical situation can always be described to
any desired level of precision with a finite number of spheres.
If greater precision is desired, a finer description with more
spheres is always possible, but at no stage is there ever an
infinite number of spheres. (014)
With this way of talking the word 'infinity' is just a loose way
of talking about an endless progression of finite descriptions.
I'll grant that Cantor's way of talking is consistent, and
I wouldn't try to stop anyone from using his terminology.
But it's possible to talk consistently about geometry and
arithmetic without any use of the word 'infinity'. (015)
John (016)
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