[FOM] naive continuum metaphysics

Bas Spitters spitters at cs.kun.nl
Wed Mar 3 02:38:50 EST 2004

On Tuesday 02 March 2004 16:14, William.Piper at colorado.edu wrote:
> Following up on Steven's question regarding the continuum as a non-set:
> I was wondering if anyone on the list has seen mathematical or
> philosophical work on the continuum being a fundamental unity? To consider
> the continuum as a pointless entity where the reals are in some sense
> indiscernable from one another was proposed to me by a friend. This person
> regards the CH as a fundamentally meaningless question and claims that we
> make the mistake of thinking of the continuum as composed of objects
> (points or reals) when it is really a single, "solid" entity. Has anyone
> seen any writing on this or perhaps written something on this themselves?

You may want to have a look at the intuitionistic literature on the continuum.
Similar questions motivated Brouwer to develop his intuitionistic mathematics. 
Heyting's book is a nice introduction, other standard reference include 
Troelstra/van Dalen and Beeson.

For a more "modern"  approach (consistent with classical mathematics) you may 
want to look at pointfree (aka pointless) topology. For instance, the work on 
formal topology (http://www.math.unipd.it/~logic/rgl/frame.html) or the 
literature on locale theory (Johnstone's Stone spaces is a standard 

Regarding CH, in intuitionistic mathematics on can look at it as follows:
It is not meanless (as your friend states), but only not formulated precisely 
enough. In intuitionistic mathematics, there are several ways of making this
precise. Some are provable, some are not.
See, Gielen, de Swart and Veldman, W, The continuum hypothesis in 
intuitionism, J. Symbolic Logic,46,1981,1,121--136

I hope this is of any help.

Bas Spitters

Department of Computer Science, University of Nijmegen.
P.O.box 9010, NL-6500 GL Nijmegen, The Netherlands.
e-mail: spitters at cs.kun.nl  Phone: +31-24-3652631. 
Fax: +31-24-3652525 Room: A5024

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