FOM: why Con(ZFC) isn't finite combinatorial; epochalness

[Stephen G Simpson]

Neil Tennant writes:
 > I believe one can understand Con(ZFC) without knowing anything about
 > Zermelo-Fraenkel set theory.

and goes on to sketch the familiar view of it as an excruciatingly
complicated statement about manipulation of strings of symbols, to be
"understood" without reference to sets or even deductions in the
predicate calculus.  Neil has a point, but clearly this notion of
"understanding" is extremely weak, much too weak to be in accordance
with normal pedagogical or psychological usage.  No human being could
truly understand Con(ZFC) in this sense; there is too much to hold in
one's consciousness.  The normal understanding of "understanding"
embraces a requirement of simplicity: one understands by boiling down
to a small number (at most 3 or 4) of key concepts on which everything
turns.  This respect for simplicity is absent from the notion of
"understanding Con(ZFC)" that Neil is playing on.  On the other hand,
the normal sense of understanding Con(ZFC) *does* respect the
requirement of simplicity, because it boils down to three key
concepts: sets, predicate calculus, consistency.

 > It seems that the foregoing reflections establish that Con(ZFC) is
 > a finite combinatorial statement, requiring no set theory for its
 > grasp.

Again, this has to be taken with a large grain of salt.  Neil is
playing on a notion of "grasp" which is not in accord with normal
usage.  The truth is that human beings grasp Con(ZFC) only by grasping
the meaning of set theory.

 > That people became interested in it and its provability or
 > non-provability within ZFC *only because of its importance for set
 > theory* is beside the point. That is just historical and
 > psychological gloss.

I hate to repeat myself, but this is *not* beside the point; it's of
the essence.  We are talking about *understanding*, a psychological
concept.  One can't understand understanding in any terms except
psychological ones.  Therefore, we can't in good conscious
characterize the set-theoretic context of that understanding as mere
gloss.

 > Having said all that, compare it now to Harvey's various finite
 > independence statements. I don't know how to *formalize* the
 > difference, but it is one that is palpably there.

Yes, that's my point.  And Neil states the palpable difference very
clearly:

 > Harvey's finite combinatorial statements seem (veridically)
 > interesting and important *as conjectures of ordinary mathematical
 > practice as soon as one understands them in their own
 > terms*. 

Note that Neil's phrase "in their own terms" refers to *combinatorial*
terms, i.e. a nexus of concepts that are familiar in a particular
well-known branch of mathematics known as finite combinatorics.  It's
reasonable to express this by saying that Harvey's statement is finite
combinatorial, in a sense that Con(ZFC) is not.  This sense of "finite
combinatorial" is in accord with normal mathematical usage.  Neil is
using language somewhat differently, but I think we agree on the
substantial point.

Now, back to the original issue.  My original formulation of why
Harvey's independence result is epochal depended on noting that the
independent statement is of a finite combinatorial nature.  Shoenfield
objected because, according to him, Con(ZFC) is also of a finite
combinatorial nature.  However, it is so in a very different sense,
and it seems to me that this difference is fatal to the objection.

Another objection raised by Shoenfield is that Harvey's work has not
yet inspired a host of imitators.  It may yet do so, but in any case
that's not a requirement for epochalness; see point 5 of Neil's
posting of 2 Aug 1998 09:57:29.

-- Steve