[FOM] "Connection set theory"

Randall Holmes m.randall.holmes at gmail.com
Thu Mar 25 11:07:47 EDT 2010

Dear colleagues,

Since my name is mentioned in the post of this title, I thought I
would make a couple of remarks.  The author contacted me and described
(somewhat informally) a criterion for comprehension.  With any formula
phi in the language of equality and membership  we associate a
multigraph whose vertices are the variables appearing in phi and which
contains an edge {x,y} for each atomic formula whose variables are
exactly x and y [note that we create an edge for each atomic formula,
not for each occurrence of an atomic formula].  We say that phi is
connected iff this multigraph is acyclic (which includes "contains no
loops", so x E x does not occur).  The comprehension axiom of the
author's connection set theory asserts that {x|phi} exists for each
connected formula phi.

I pointed out to the author that a connected formula is stratified,
and so connected comprehension is consistent with weak extensionality
(nonempty sets with the same elements are the same) because this is a
subtheory of NFU, and connected comprehension with strong
extensionality is a subtheory of NF.

I absolutely do not agree with the author that the connectedness
criterion for comprehension is in any way obviously "safe":  the way
we see that it is consistent exploits the consistency of stratified
comprehension, and history shows that the criterion of stratified
comprehension is not *obviously* safe, though it *is* safe.  It took
32 years from 1937 (proposal of this criterion) to 1969 (consistency
proof).  Nor is it particularly appealing:  there is no semantic
intuition behind this proposal at all.

There is a technical question about this comprehension criterion:  is
it equivalent to full stratified comprehension?  My guess is that the
answer is negative:  in particular I do not think that the subset
relation can be shown to be a set, but I have not as yet solved this
puzzle.  I would conjecture that the theory (even with strong
extensionality) is consistent and very weak.  I think this question is
mostly of interest to those few of us who concern ourselves with
subsystems of NF.

Sincerely, Randall Holmes

Any opinions expressed above are not the
official opinions of any person or institution.

More information about the FOM mailing list