Paul Taylor, who has often contributed to MO, once posted this to the categories mailing list; the original thread can be found here:

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From: Paul Taylor

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Subject: categories: Is Zermelo-Fraenkel set theory inconsistent?

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IS ZERMELO-FRAENKEL SET THEORY INCONSISTENT?

At the end of this message is a sketch of an argument that leads to the conclusion that Zermelo-Fraenkel set theory is inconsistent.

The impact on mathematics is not as devastating as the incautious observer might suppose. Recall that ZERMELO set theory (1908), which is essentially equivalent to the categorists' notion of ELEMENTARY TOPOS with natural numbers and the axiom of choice, is adequate for most of the purposes of mathematics, though not, as I shall try to explain, logic (and theoretical computer science).

ZERMELO-FRAENKEL set theory is the extension of this system by the axiom-scheme of REPLACEMENT, which was first formulated by Adolf (later Abraham) Fraenkel, Nels Lennes and Thoralf Skolem in 1922, although Dimitry Mirimanoff already had something of the idea in 1917. Notice that this is some two decades after the appearance of the famous "antinomies" of set theory, so presumably the set theorists' guard had dropped by that time, and they had begun again to assert megalomaniac axioms. On the other hand, it is a decade before the second generation of paradoxical results, Godel's incompleteness theorem and Turing's unsolvability of the Halting Problem.

Whenever I see set theory books in a library or bookshop, I turn to the index to find out what they have to say about Replacement. Usually there is some trivial result, such as the existence of what categorists call image factorisation, that could have been proved from Zermelo's axioms with a little more facility in set-theoretic constructions.

The basic use of Replacement, that you will find in the better set theory books, is the recursive construction of sets (in substance -- types or objects to type-theorists and categorists -- rather than their names).

For example, Mostowski's theorem states that every well founded extensional binary relation < is isomorphic to the membership relation for a unique transitive set. This is found, recursively, by means of the formula f(x) = { f(y) | y < x }, which also provides the extensional reflection (quotient) of any well founded relation. In fact the latter result (where the quotient relation is merely another relation, rather than a membership relation) can be proved using the topos or Zermelo axioms alone, and not Replacement [1], although there are categorical generalisations of this that certainly do need Replacement.

Richard Montague [2] proved a result that should have been taken as a warning of the perilous nature of Replacement, though I suspect that Montague's personal eccentricities may have been the reason why he was ignored. ZF can prove the consistency, not only of Zermelo set theory (Z) itself, but also of Z extended by any single theorem of ZF.

Adrian Mathias has claimed [3] that Bourbaki was "ignorant" of Replacement, ie that it did not occur in "Theorie des Ensembles" [4]. Although Bourbaki is hardly very clear on this matter, it does include a version of Replacement in its axioms, indeed one that is in widespread use in category theory and other parts of mathematics, namely that one can form the UNION of any SET-INDEXED FAMILY of SETS.

One application of N-indexed unions in theoretical computer science is Scott's "D-infinity" construction of models of the untyped lambda calculus. Starting from any domain D0=D, one may form its function-space D1=(D0->D0), and similarly D2=(D1->D1), etc., linking these together with embedding- projection pairs. If D was one of the examples of L-domains, having a pair of elements with infinitely many minimal upper bounds, then one can show (classically) that D-infinity has the cardinality of a model of Zermelo set theory, so need not exist within such a model unless it also satisfies Replacement.

These two ways of seeing Replacement have a common theme: we use N-indexed or transfinite unions to unfold a free(ish) model of one logic within a model of another.

Having seen this in the context of a messy domain-theoretic construction, we may think in a more disciplined way about free models of the lambda calculus, the topos axioms, etc. In fact, there is no difficulty in constructing these models, as they are merely TERM ALGEBRAS. The problem lies in proving that the term algebra has the universal (initiality) property that qualifies it as "free":

Let S be the universe (the category of ZF-sets, for example) and F the term algebra (internal to S) for the logic L. Suppose that S itself is a model of L.

Then there is a unique interpretation functor []:F->S that takes each syntactic operation of F (eg prod(a,b)) to the semantics ([a] x [b]) in S.

It is merely unique up to unique isomorphism if the L-structure in S is defined by universal properties rather than being chosen.

This initiality property may also be expressed type-theoretically. Per Martin-Lof [5] introduced objects with such a property, called UNIVERSES, observing the analogy with Replacement. This point of view stresses that the above property is a RECURSION SCHEME.

Let me explain how I came to realise that the existence of []:F->S depends, in general, on Replacement.

There is an amazingly simple but incredibly powerful argument, due to Peter Freyd and known variously as (Artin-Wraith) glu(e)ing, sconing, the Freyd cover, logical relations and other names. It is based on some very elementary categorical investigations of a certain comma category involving F and S. This argument has been developed rather a long way (the most recent paper that I know of is [6]), and we are pretty close to having a purely categorical proof of the strong normalisation theorem for lambda calculi that, unlike the syntactic proofs, is completely generic with regard to the calculus in question.

Freyd originally showed that the terminal object (1) of the free topos (F) is projective, and more generally the "global sections functor" F(1,-) : F -> S preserves colimits. In particular, it preserves the initial object (0), which is categorical jargon for saying that S proves the consistency of F, because the S-set of F-morphisms 1->0 is the initial (empty) S-set.

I found this suspicious, because the punch-line of Andre Joyal's 1973 (but as yet unpublished and unavailable) categorical proof of Godel's incompleteness theorem is that such a functor F(1,-) : F -> S does not preserve the initial object.

The more careful amongst categorists ought also to be suspicious when I speak of "a functor F(1,-) or [] : F -> S" where F is an INTERNAL category in S. The meaning that we must give to this phrase is that it is "syntactic sugar" for a certain FIBRATION p: V -> F, where V is also an internal category and p and internal functor in S.

This brings us back to the relationship between Replacement as a recursive construction of objects and Replacement as infinitary colimits: "p: V -> F" is the colimit (in a 2-category whose objects are fibrations) of a recursively defined diagram vaguely similar to that which gives Scott's D-infinity.

I have come to the conclusion that attempts to define "colimits" such as this are inherently circular: what, after all, does it mean to have a "cocone" to test such an alleged colimit?

My categorical formulation of Replacement speaks about fibrations and smaller colimits defined internally in the style of Benabou. This is to be found in the final section (9.5) of my book [7], Section 7.7 of which also gives an account of Freyd's gluing construction.

This book is officially due to be published in mid-May, but it is already in stock (and I have my own copy in front of me), and is available direct from the publishers at 50 pounds (inclusive of overland postage and packing). Please contact Richard Knott, email: rknott@cup.cam.ac.uk fax: +44 1223 315 052 tel: +44 1223 325 916 (but other methods are preferable) snail: Cambridge University Press, The Edinburgh Building, Shaftesbury Road, Cambridge, CB2 2RU, UK with your address and credit card number. (2.50 pounds extra for airmail.)

Having seen that Replacement provides a UNIFORM way of proving consistency of any fragment of logic, we come at last to the inconsistency argument:

Let L(0) be Zermelo set theory (or the axioms for an elementary topos).

For each n, let L(n+1) be L(n) plus as much of the axiom-scheme of replacement as is needed to justify the gluing construction that shows that

L(n+1) |- ``L(n) is consistent.''

Now let L(infinity) be the union of L(n) over n:N.

If L(infinity) |- false then L(n) |- false for some n.

But L(infinity) |- ``L(n) is consistent,''

so L(infinity) proves its OWN consistency, contradicting Godel's theorem.

However, L(infinity) has a standard non-trivial interpretation in Zermelo--Fraenkel set theory, which is therefore inconsistent.

[1] Paul Taylor, Intuitionistic Sets and Ordinals, JSL 61 (1996) 705-44

[2] Richard Montague, Fraenkel's Addition to the Axioms of Zermelo, pp 91--114 of Bar-Hillel, et al., eds., Essays on the Foundations of Mathematics, Magnes Press, Hebrew University, 1966 (distributed by Oxford University Press).

[3] Adrian Mathias, The Ignorance of Bourbaki, Mathematical Intelligencer, 14 (1992) 4-13.

[4] Nicolas Bourbaki, Elements de Mathematique XXII: Theories des Ensembles, Livre I, Structures, Hermann, 1957 (English translation 1968).

[5] Per Martin-Lof, An Intuitionistic Theory of Types: Predicative part, pp 73--118 in Rose and Sheperdson, eds., Logic Colloquium '73, North-Holland, Studies in Logic and the Foundations of Mathematics #80, 1975

[6] Djordje Cubric, Peter Dybjer and Philip Scott, Normalisation and the Yoneda Embedding, MSCS 8 (1998) 153--192.

[7] Paul Taylor, Practical Foundations of Mathematics, Cambridge University Press, Cambridge Studies in Advanced Mathematics #59, xii+572pp, 1999.

http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/

Paul Taylor 19990401

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