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Functional Set Theory

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Functional Set Theory FUNCTIONAL SET THEORY Michael Barr∗ Department of Mathematics and Statistics McGill University 805 Sherbrooke St., W Montreal, Quebec Canada H3A 2K6 Email: [email protected] June 11, 1999 1 Introduction and Moss paper is to describe a novel kind of non-well- founded set theory in which such infinite chains are A recent article in The Intelligencer described in some allowed. The purpose of this note is to suggest that detail the theory of non-well-founded sets for the pur- the whole enterprise is wrong-headed. pose of being able to solve freely equations of the sort It is clear that the \intended solution" to the equa- X = A × X with A fixed ([Barwise & Moss, 1991]). tion X = A × X is that X should be the set of all Of course, the empty set is a solution, but what was sequences wanted was a non-empty solution (in fact, the largest solution, if such exists). If X were a non-empty so- ha0; a1; : : : ; an;:::i lution and x 2 X, then we would have x = ha1; x1i for some a1 2 A and x1 2 X. Here I use h−; −i to for a0; a1; : : : ; an;::: 2 A. The reason that classical denote ordered pairs. What happens next depends to set theory cannot provide this solution is not that this some extent on the definition of ordered pair, but let us set of sequences does not exist as that for the standard take the common definition that hu; vi = ffug; fu; vgg definition of product the set Seq(A) does not satisfy which is a set with one or two elements, depending on Seq(A) = A × Seq(A). Even if there were some alter- whether or not u = v. It seems a little complicated, but nate definition of product that allowed this definition it works and I believe that the same problem would oc- to be satisfied, there would probably come along some cur no matter what definition of ordered pair is taken. similar equation for which a solution was wanted and Let us write u 22 U to indicate that there is a for which the new definition was inadequate. 2 t with u 2 t and t 2 U. For example, u 2 hu; vi The theory of non-well-founded sets solves this 2 and v 2 hu; vi. Now getting back to the equation problem and all similar problems. It is unfortunate x = ha1; x1i with a1 2 A and x1 2 X, we have that such solutions exist, for their main effect is to 2 that x1 2 x. Since x1 2 X, it can be written as avoid giving serious consideration to the real problem: x1 = ha2; x2i with a2 2 A and x2 2 X. This implies the irrelevance of actual elements in mathematics. that x 22 x . Continuing in this way, we have a se- 2 1 This is perhaps best illustrated by a two-person quence of elements x 2 X such that n game I heard described by someone recently (I have 2 2 2 2 unfortunately forgotten the source): Player A begins ··· xn 2 xn−1 2 · · · 2 x1 2 x by choosing a construction of the complex numbers. This infinitely descending 2-chain is banned (by a spe- Call it C. Player B then chooses an element z0 2 C. cific axiom of \well-foundedness") from standard treat- Then A chooses a z1 2 z0. B responds by choosing ments of set theory and the main point of the Barwise an element z2 2 z1. They alternate this way until the ∗In the preparation of this paper, I have been assisted by grants from the NSERC of Canada and the FCAR du Qu´ebec. 1 player unable to choose an element loses. By the well- placement axiom scheme to save my life, I would have foundedness axiom, this is a finite game. By making perished in ignorance. Then a colleague went on leave choices in the construction of C that are clearly irrel- and I was asked to teach a course on set theory. I did evant from a mathematical point of view, it is clearly teach the course and in the process a curious thing hap- possible to describe a myriad of different sets, each of pened. I lost all respect for set theory as a foundation which might function as \the" complex numbers. for mathematics. The reason is simple. In virtually There are two (at least!) rather differing views on all branches of mathematics we define various kinds the relation of set theory to mathematics (from which, of structures and then define some kind of admissi- for the purposes of this paragraph, I exclude set the- ble homomorphisms as those that preserve the struc- ory itself). The first, which I held and supposed that ture. This is true everywhere but in set theory. The mathematicians held quite generally, was that we took elaborate structure exists, but we pay absolutely no sets seriously and imagined that all mathematical ob- attention to the functions that preserve it. Every non- jects were sets, that the number 3 was the set f0; 1; 2g, empty set has its elements and the elements, unless that the sine function, say, was a rather complicated empty, have elements and they, when non-empty, have set and so on. A different view (which appears, on elements in turn. We get this elaborate tree and the the basis of a limited sample, to be the common one), foundation axiom requires that whenever you follow a is that the main thing set theory does is to show that downward path through this tree, you eventually come there are models of these things inside these axioms. In up empty. particular, if set theory is consistent, then so are the What would it mean for a function to preserve this constructions that could be made using them. (How- structure? For a function f: X −! Y , not only should f ever, it goes beyond consistency, since there are lots of map elements of X to elements of Y , but also if x 2 X, things that are consistent with the axioms that we do then f should map the elements of x to elements of not assume because there is no model of them inside f(x) and so on. For most sets X, the only function the standard axioms.) One thing we agree on is that, X −! X that does this is the identity! Of course, whatever they believe, mathematicians do not act as all this sounds silly because while one is interested in though they took seriously the idea that the zeta func- the elements of a set, one is virtually never interested tion is nothing but a rather complicated set. For the in the elements of the elements. To take one exam- purposes of this article, it doesn't matter which view ple, which I take from a recent paper of Colin McLarty you hold. The foundations described here are in any [preprint, 1991], suppose you consider two possible def- case much closer to the actual practice of mathematics. initions of the natural numbers. In the first (common) As an example, consider the question of what is an definition, 0 is defined to be the empty set and we de- ordered pair. One answer is that hu; vi = ffug; fu; vgg fine n = f0; 1; : : : ; n − 1g so that n has n elements. In as described above. Other definitions are possible, but the second definition, 0 is still defined to be the empty let us ignore that. On the one view, this is what an set, but n = fn − 1g. The two definitions of natural ordered pair is. On the second view, that is no more numbers are quite different. Is the number theory that than a model to show that the specification of ordered results different? Assuming it isn't, why are we con- pair by hu; vi = hu0; v0i implies u = u0 and v = v0 is cerned with the set theory at all? See also [McLarty, to consistent. One thing for certain is that the specifica- appear] in which he shows how to simplify and gener- tion is the only property of ordered pair that is ever alize Barwise and Etchemendy's representation of the used. Liar paradox, precisely forgetting about the identity of If one takes sets seriously, then to solve an equa- elements. tion X = A × X requires demonstrating the existence The problem in all these cases is not so much that of a set with that property, which is impossible with sets have elements as that the elements are sets that well-founded sets. If one doesn't take them seriously, have elements. There is a serious reason for this. then the only issue is the consistency of the assump- Many constructions, for example, products and quo- tion. The equation X = the set of subsets of X can tient structures, are carried out by building sets whose easily be shown (by Cantor's diagonal argument) to be elements are sets. From there it is a short distance to inconsistent with any reasonable idea of sets. assuming that all elements are sets, since that certainly Until a few years ago, I knew as much about set gives a parsimonious foundation. theory as the average mathematician, which is to say The set theory of the Zermelo-Fraenkel, G¨odel- virtually nothing. Had I been asked to state the re- Bernays and similar axioms is reminiscent of defining 2 vectors as n-tuples of scalars in that it introduces the y 2 A × X implies that there are unique a 2 A and basis as part of the strucutre.
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