FOM: The consistency of ZF
J.P.Mayberry at bristol.ac.uk
Sat Mar 20 05:28:19 EST 1999
Why do we believe the formal system ZF to be consistent? It
seems to me that there are two kinds of reasons, one kind mathematical,
the other sociological, if I may put it that way.
The mathematical reasons arise out of our acceptance of
Zermelo's careful analysis of what a set is. On that analysis a set is,
roughly, a collection (class, multitude, etc) that is not too large to
admit of mathematical determination. Put that way it is rather vague,
but when you get down to looking at Zermelo's axioms, you can see that
they all make sense on that interpretation, and, what is more, make it
much clearer what is meant by "not too large" in this context. The set
of natural numbers, though infinite, is small relative to other
infinite sets (Cantor taught us to see things that way); if the class S
is not too large then the class of all its subclasses is not too large
either. Etc. In fact these two axioms (Infinity and Power Set) are the
controversial ones. In particular, the axioms of Comprehension and
Replacement are utterly obvious from this point of view.
To get from considerations of this kind to the formal
consistency of ZF we simply have to convince ourselves that ZF is just
a formalisation of the axioms that Zermelo arrived at on his analysis.
But here we run into a difficulty, and it arises in an unexpected
quarter. The problem is with Replacement: we are convinced that the
image of a set under a function is a set - how could the image of S
under a well-defined function be "too large" if S itself is not "too
large"; but what gives us the right to suppose that we can define a
function by means of a complicated formula of ZF which involves
intricate nestings of unbounded quantifiers ranging over the entire
universe of sets? After all, that universe is too large to be a set -
it is, as Cantor said, *absolutely* infinite.
We are already familiar with this kind of difficulty in
connection with mathematical induction over the natural numbers.
Everyone believes induction to be valid when applied to a well-defined
property. But does a complicated formula of PA involving intricate
nestings of quantifiers ranging over the set of natural numbers
determine such a property? You have to see that the difficulty lies,
not with induction, but with its embodiment as a schema in PA:
otherwise you won't understand the significance, and subtlety, of
Gentzen's consistency proof. After all, Gentzen's proof employs
induction, transfinite induction up to epsilon zero: but that induction
is with respect to a property defined by a *quantifier free* formula.
And notice: the "impredicativity" of Replacement in ZF is, prima facie,
more serious than the "impredicativity" of induction in PA. For each
instance of Replacement in ZF is "enlarging" the universe over which
its own quantifiers range.
There is thus a gap between our mathematically rigourous, but
non-formal conception of set based on Zermelo's analysis, and the
formal theory ZF. Even if we accept Zermelo's analysis of set, there is
still the possibility that the formal theory ZF may be inconsistent.
But it is here that the "sociological" considerations I referred to
above come into play. For some of the best mathematical minds of the
20th century have worked - are still working - with ZF and, indeed,
with extremely powerful extensions of ZF, without having encountered a
contradiction. If there is a contradiction there it is going to be very
hard to winkle out. So we all believe in the consistency of ZF; but our
reasons for doing so are not entirely mathematical.
School of Mathematics
University of Bristol
J.P.Mayberry at bristol.ac.uk
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