[FOM] RE: FOM New Axioms

Todd Wilson twilson at csufresno.edu
Sat Jun 28 18:07:10 EDT 2003

On Fri, 27 Jun 2003, Matt Insall wrote:
> [Ralph Hartley]
> Does this principle [vaguely: when possible, sets behave as do
> finite sets] always give unique results?
> [Ralph Hartley]
> I can't off hand think of a statement that is true/provable for all finite
> sets which, if extended to all sets, implies CH or ~Choice, but can you
> prove there isn't one?
> [Ralph Hartley]
> How hard is it to find a pair of statements that are both true for all
> finite sets, but which can't both be true for all sets (including infinite
> ones)?
> [Ralph Hartley]
> How many statements are there that are provable/true for all finite sets,
> but who's extension to all infinite sets is independent of the axioms?

Some trivial answers to these questions can be constructed according
to the following pattern:  if S is any statement in the language of
set theory, then the statement

    if there exists an infinite (or "inductive") set, then S

is (vacuously) true in a universe of finite sets and equivalent to S
in a universe with infinite sets.

When I was a graduate student in the early 90s, I toyed with the idea
of finding a useful, consistent fragment of the inconsistent system
that has full comprehension as its only axiom (scheme).  The goal was,
not unlike with Quine's system NF, to find a principled way of
choosing comprehension axioms so that (1) all of the usual set
existence axioms -- empty set, pair, union, powerset -- would be
present or derivable, and (2) no inconsistencies would result.

One approach, following a doctrine of size, was to take only those
instances of comprehension with parameters that would produce
sets/classes whose size was "bounded" in terms of the sizes of the
parameters.  One way of measuring this boundedness was to check
whether the comprehension axiom, when applied to a universe of
hereditarily finite sets, produced a hereditarily finite set.
However, this didn't work out, as my advisor Dana Scott pointed out to
me, for reasons similar to the above, and being somewhat pressed for
time, I didn't purse the idea any further.

I wonder now, however, reading the exchange between Matt Insall and
Ralph Hartley, whether this idea has been explored elsewhere.  So let
me ask:  has anyone (besides Quine) proposed a system of set theory
whose axioms consist entirely of instances of full comprehension?

Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh

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