[FOM] "Connection set theory"

[Randall Holmes]

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.