FOM: Understanding Con(ZFC): some problems
neilt at mercutio.cohums.ohio-state.edu
Sun Aug 23 13:18:28 EDT 1998
Steve maintains that I was using "grasp" (of meaning), or
"understanding", in some sort of non-standard sense when I claimed
(with an appropriate argument to back the claim) that one could
understand the statement called Con(ZFC) without having to know
anything about set theory. He claims that Con(ZFC) is too long to hold
in one's consciousness, as though this is a fatal objection to one's
claim to be able to understand it.
But if that objection is fatal, then it is equally so to the claim
that one can understand Con(ZFC) provided only that one know some set
theory, and that one take Con(ZFC) to be making the claim that the
theory ZFC is consistent.
For, in primitive notation, Con(ZFC) is just as long for the
set-theory-understander as it would be for the
set-theory-ignoramus. Both these thinkers would have the same problem
of trying to "hold it in consciousness".
If the set-theory-understander can get around the problem of excessive
length by "chunking" the sentence Con(ZFC) by means of appropriate
definitions (crucially exploiting the notion of proof etc., via G"odel
numbering), then the set-theory-ignoramus could employ the same
strategy but by different chunking. He/she could use a tree of chunking
definitions so that Con(ZFC) was rendered more manageable in terms of
certain compiled concepts. These alternative chunking definitions
need not advert at all to notions of proof, G"odel-numbering etc.
For, after all, Con(ZFC) is of the form "not: there exists x such that
F(x)" where F(x) is just built up out of zero, successor, plus, times
and the logical operations of connection and bounded numerical
quantification. One could *arbitrarily* boil such a statement down to
one of manageable length, by means of appropriate abbreviatory
The issue of length, and the contingent ability to hold a sentence in
its entirety in one's consciousness, seems to me to be a red
herring. It is common, in the theory of meaning, to appeal to the
notion of being able, *in principle*, to do something/recognize
something/carry out some procedure, etc., without regard to its
feasibility (in terms of space- or time-requirements).
Moreover, Con(ZFC) as I have just described it---in terms of its
primitive vocabulary---*has* to admit of a reading that is based
*entirely on those primitive expressions, and the way they are put
together in the sentence*. To deny this is to deny the
principle of semantic compositionality, which goes back to Frege.
Interestingly, the reading that Steve prefers for Con(ZFC)---the one
that adverts to proof etc.---depends on one's knowing the result that
certain recursive notions (such as "x is a proof of y") are
representable in a sufficiently strong theory of arithmetic. Only
then can we appreciate the sentence as effecting the kind of
"syntactic reference" that has to be involved in order for us to be
able to take it as making the claim of consistency. But such
theoretical insight should not be a *presupposition* for absolutely
every licit kind of understanding. The person who simply grasps the
meanings of zero, successor, plus, times, and the logical operators is
*already* in a position, in principle, to grasp the meaning of the long
sentence called Con(ZFC). It would be wrong to demand of him/her a
knowledge of numeralwise representability of recursive functions as a
prerequisite for *semantic grasp*.
Thus, I still maintain that the meaning of Con(ZFC) that is graspable
simply on the basis of one's grasp of its primitive expressions is the
proper one with which to contrast the meanings of other independent
statements, in order to judge the importance or otherwise of the
respective independence results.
If Steve denies this, then he is buying into a strange combination of
semantic dogmas. For, first, he would be introducing a psychologistic
element with the notion of "holding a sentence in one's
consciousness". Secondly, he would be advocating an extreme
*theoretical-cum-semantic* holism, by insisting that one would have to
know the representability theorem in order to make sense of a sentence
built out of the primitive expressions of first-order arithmetic.
Thus Steve would be going against Frege, one of the greatest
foundationalists, by denying compositionality and insisting on
holism. And the denial of compositionality would be bringing with it
the extra twist that one could claim an understanding of syntactic
objects only when they were of feasible length. This would raise the
spectre of finitism.
It would be an extraordinary philosophical combination: Quinean holism
(presumably with quietism over the choice of classical logic as the
correct logic), along with a strand of finitism that would be chafing
to destabilize the classicism to a point way beyond intuitionism. I
believe such a position to be extremely unstable, but justifying that
belief would take a much longer philosophical argument than I would
have space for here.
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