[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.
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