[FOM] The lure of the infinite

[Isaac Malitz]

Quoting  Martin Davis <martin at eipye.com>

> Warnings about the dangers and contradictions lurking in reasoning about
> the infinite have abounded since the ancient Greeks. ...
>
> In a recent post, Weaver reminds us that the Russell-Cantor antinomies
> concerning the set-theoretic hierarchy are still with us. How can we have 
> a
> notion of set such that one cannot form the set of all the entities that
> fall under it? Workers in f.o.m. are far from having a good answer to this
> question. Set theorists may speak of proper classes, but, as one of them
> asked me not so long ago: am I really comfortable with that? Aren't proper
> classes really just sets in disguise?
>

These issues (and some related ones) can be defused by accepting the view
that the only "true" concept of a set is a very primitive one; and that what
mathematicians do is to work with quite radical extrapolations of that
original concept. In what follows, I'll try to explain:

[1] By the Ordinary Conception of a Set ("OC Set") I mean simply the
ordinary, "man-on-the-street", non-mathematical conception of a
set.

I think that OC Set only allows for small finite sets,
consisting of things "right in front of one's nose" that are
somehow related to one another. For instance: A set of dishes; a set of
data files in my computer; my dog's set of teeth; and so on.

[2] I think that strictly speaking, anything which goes beyond the
above guidelines is not the Ordinary Conception of a Set! For
instance: The set of all of my business records; the set of all
Social Security Numbers assigned in the United States prior to
05/12/94; the set of all atoms in the universe as of a certain
moment; the null set; the set of all positive whole numbers less
than 1,000,000 (By a "whole number" I mean an actual number, not
the set-theoretical construct.).

[3] Now obviously, all of the examples in [2] are in some sense
legitimate discourse. How are we to explain this? I think the
answer is that *it is part of the concept of OC Set that it is
"extendable"* . Built into the concept is "permission" to
"stretch" the concept. So, even though the set of positive whole
numbers less than 1,000,000 is not "really" a set, it is okay to
stretch the concept and speak of those numbers as a set.


[4] With the above observations as a background, I will now attempt a
philosophical characterization of what a set is:

     PCS1: The Ordinary Conception of a Set (OC Set) is as
     follows: An OC Set is required to be a small number of
     individuals which in some sense are related - where it is
     natural to speak of them as an aggregate. There should be
     some principle, fact, or consideration which identifies
     these individuals and explains how they are related. The
     individuals should be clearly identifiable, previously
     established entities. [Possibly some additional
     constraints.]

     PCS2: It is permissible to extend the concept of a set
     beyond the constraints of PCS1. This is done by relaxing
     various requirements in PCS1. Whenever this is done, a
     *justification* should be supplied for the appropriate
     relaxation of requirements.


(I think that the *aggregation requirement* in PCS1 is quite
strong. The most natural examples of OC Sets seem to be where it
is about as natural to speak of the aggregate as it is of the
components. E.g., "Judging from the condition of my shoe, your
dog must have a nice set of teeth.")

[5] Using PCS1 and PCS2 one can proceed to formulate many
*Extended Conceptions* of a set. Some conceptions seem quite
natural - e.g. the iterative conception, or perhaps the conception
that underlies NF (as per Randall Holmes, Thomas Forster).
Other conceptions seem to me rather artificial (e.g. Church's
Set Theory with a Universal Set); but maybe the mathematics is interesting.

[6] Strictly speaking, I think any Extended Conception lacks
the epistemological certainty/clarity of OC Set. So when it turns
out that some Extended Conception entails antinomies or other
disturbing consequences, that's not something to be alarmed
about. Extrapolation of OC Set is a fascinating but also risky exercise.

[7] Applying the above to issues raised by Prof. Davis:

      [a] I think that any conception of an infinite set is shaky
      from an epistemological point of view.

      [b] The same re the Universal Set (set of all sets)

      [c] Once the fundamental role of OC Set is understood,
      then the antinomies of set theory are resolved (from an
      epistemological point of view). I.e. if the Unrestricted
      Comprehension Principle is considered not-obvious,
      then strictly speaking there is no longer an "antinomy".

      [d] When mathematicians are working with Extended Conceptions
      of OC Set,  they are involved in a combination of
      straight ("technical") mathematics  and various
      degrees of philosophizing/conceptualizing. But all of this
      activity entails some risk (of antinomy or other disturbing 
consequences).


Isaac (Richard) Malitz, Ph.D.
Los Angeles, CA
imalitz at rdic.com
[I was a Church student in the mid-1970s.
Dissertation on topological set theory.]