FOM: The consistency of ZF

John Mayberry J.P.Mayberry at
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.

John Mayberry
School of Mathematics
University of Bristol

John Mayberry
J.P.Mayberry at

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