Date: Tue, 10 Nov 1998 18:53:15 +0100 (MET) From: Perez Garcia Lucia Subject: categories: Gödel and category theory I am interested in the foundations of mathematics -more concretely, in the claim that category theory can serve as a superior substitute for set theory in the foundational landscape. In this context, I would like to point out a footnote which appears in 'What is Cantor's Continuum Problem?', written by Kurt G?del in 1947, revised and expanded in 1964, and finally published in Benacerraf P. and Putnam H. (eds.) 1983: Philosophy of Mathematics. Selected Readings, Cambridge University Press, pp. 470-485. It reads as follows: It must be admitted that the spirit of the modern abstract disciplines of mathematics, in particular of the theory of categories, transcends this concept of set*, as becomes apparent, e.g., by the self-applicability of categories (see MacLane, 1961**). It does not seem however, that anything is lost from the mathematical content of the theory if categories of different levels are distinguished. If there exist mathematically interesting proofs that would not go through under this interpretation, then the paradoxes of set theory would become a serious problem for mathematics. *(the concept of set G?del was referring to is the iterative one). **(MacLane, S. 1961. "Locally Small Categories and the Foundations of Set Theory". In Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics (Warsaw, 1959). London and N.Y., Pergamon Press). I need some help to grasp the following questions: - In what sense the self-applicability of categories transcends the concept of set?. (It is obvious that categories transcend the concept of well- founded set but, what's the matter with non-well-founded sets?. - In what sense do you think G?del proposed distinguishing different levels of categories?. Would it be possible that G?del was thinking of something like type theory?. - Do you agree with G?del's intuition that nothing would be lost with such a distinction?. - Finally, in the last lines of the note G?del seems to suggest a research programme for category theory as an alternative foundation of mathematics. To what extent has it been carried out?. Thanks for your help. Regards, Luc?a P?rez Dpt.L?gica y Filosof?a de la Ciencia University of Valencia -Spain- -- *********************************************** lperez *********************************************** Date: Wed, 11 Nov 1998 10:03:10 -0500 (EST) From: Michael Barr Subject: categories: Re: Gödel and category theory I should really let people with more interest in foundations field this question, but fools rush in ... On Tue, 10 Nov 1998, Perez Garcia Lucia wrote: > > I am interested in the foundations of mathematics -more concretely, > in the claim that category theory can serve as a superior substitute > for set theory in the foundational landscape. In this context, I would > like to point out a footnote which appears in 'What is Cantor's > Continuum Problem?', written by Kurt G?del in 1947, revised and expanded > in 1964, and finally published in Benacerraf P. and Putnam H. (eds.) 1983: > Philosophy of Mathematics. Selected Readings, Cambridge University Press, > pp. 470-485. It reads as follows: > > It must be admitted that the spirit of the modern abstract disciplines > of mathematics, in particular of the theory of categories, transcends > this concept of set*, as becomes apparent, e.g., by the self-applicability > of categories (see MacLane, 1961**). It does not seem however, that > anything is lost from the mathematical content of the theory if categories > of different levels are distinguished. If there exist mathematically > interesting proofs that would not go through under this interpretation, > then the paradoxes of set theory would become a serious problem for > mathematics. > *(the concept of set G?del was referring to is the iterative > one). > **(MacLane, S. 1961. "Locally Small Categories and the > Foundations of Set Theory". In Infinitistic Methods, > Proceedings of the Symposium on Foundations of Mathematics > (Warsaw, 1959). London and N.Y., Pergamon Press). > > I need some help to grasp the following questions: > > - In what sense the self-applicability of categories transcends the concept > of set?. (It is obvious that categories transcend the concept of well- > founded set but, what's the matter with non-well-founded sets?. > I will pass on this one. As far as I know, using category theory as foundations gives an equally powerful, but not more powerful, foundation. But I would say the same about non-well-founded set theory. Basically, it is a matter of convenience and, perhaps, coherence. > - In what sense do you think G?del proposed distinguishing different levels > of categories?. Would it be possible that G?del was thinking of something > like type theory?. > I would assume that is what he meant. > - Do you agree with G?del's intuition that nothing would be lost with such > a distinction?. > In a word: no. > - Finally, in the last lines of the note G?del seems to suggest a research > programme for category theory as an alternative foundation of mathematics. > To what extent has it been carried out?. > Quite a bit; it is called elementary topos theory. I want to add something here. I have taught a course in set theory (twice, actually). I didn't much enjoy it, so perhaps I am prejudiced. But I have a specific complaint. In all the fields of mathematics that I have worked with (all the axiomatic fields, I should say), structures are defined and then functions that preserve those structures. The structure of a set is that of an epsilon tree, but this structure is ignored when it comes to defining functions. A good thing too, since the only functions that preserve that structure are inclusions of subsets. And the only endomorphism of a set is the identity. Believing that theory should follow practice, I am unhappy with the standard foundations that build this elaborate structure only to ignore it. When categories are used as foundations, then the undefined terms are object, arrow, domain, codomain, identity, and the relation (partial function) of composition. They are all used regularly in category theory; they are not just there to give a formal foundation. And functors are exactly what preserve these things. > > Thanks for your help. > > Regards, > > Luc?a P?rez > Dpt.L?gica y Filosof?a de la Ciencia > University of Valencia -Spain- > > -- > *********************************************** > lperez > *********************************************** > > Date: Wed, 11 Nov 1998 16:41:48 -0500 (EST) From: cxm7@po.cwru.edu (Colin McLarty) Subject: categories: Re: Gödel and category theory Perez Garcia Lucia wrote, among other things: >- In what sense the self-applicability of categories transcends the concept > of set?. (It is obvious that categories transcend the concept of well- > founded set but, what's the matter with non-well-founded sets?. Well-founding is an irrelevant detail. Take any non-wellfounded set theory which includes the axiom of choice, such as Aczel's AFA. Then every set is isomorphic to an ordinal, that is to a well-founded set. Since categorical methods are all isomorphism invariant, any categorical structure available in this set theory is also available in well-founded sets. I have discussed this in an article "Anti-foundation and self-reference" Journal of Philosophical Logic 22 (1993) 19-28. There is no real chance that abandoning the axiom of choice will help either--say by adopting AFA without Axiom of Choice. Rather, the apparent issue is existence of a universal set--a set of all sets, so that you make a category of all categories. If you want to use membership based set theory this will require non-wellfounding, but again the details of membership and wellfounding are irrelevant. Anyway, the problem here is that functions are hard to work with in set theory with a universal set. I have shown that in any such set theory meeting a few weak conditions there is a category of all categories, and it is not cartesian closed. The result is clear from the more particular case "Failure of cartesian closedness in NF" Journal of Symbolic Logic57 (1992) 555-56. Working with such a poor 'category of all categories' is much more difficult than just doing without. I think a more promising approach is to use Benabou's theory of fibrations and definability as in Benabou J. (1985). "Fibered categories and the foundations of naive category theory". Journal of Symbolic Logic 50, 10-37. I have discussed this briefly in "Category theory: Applications to the foundations of mathematics" Routledge Encyclopedia of Philosophy (1998); and in "Axiomatizing a category of categories" Journal of Symbolic Logic56 (1991) 1243-60. I see no good arguments that there SHOULD be a genuine "category of all categories" in any strong sense. But it seems an interesting question. >- In what sense do you think G?del proposed distinguishing different levels > of categories?. Would it be possible that G?del was thinking of something > like type theory?. More likely he was thinking of Eilenberg and Mac Lane's use of Goedel-Bernay's set theory as a foundation in "The general theory of natural equivalences", so there are set categories and class categories. To study Goedel's claim here, you should look at any of Mac Lane's papers on foundations that Goedel might have seen by this time. Maybe the foundational parts of "The general theory of natural equivalences" are all he could have seen, I don't know. Then it would be good to know what people around Princeton were saying about category theory at this time--and that might be very hard to find out. Colin Date: Thu, 12 Nov 1998 00:33:47 +0100 From: aurelio carboni Subject: categories: Godel AS for the Perez calling for references, I found quite surprising that nobody quoted Lawvere's work on the subject. Carboni. Date: Mon, 16 Nov 1998 15:07:52 -0500 (EST) From: F W Lawvere Subject: categories: re: Sets Conceptualizing and axiomatizing Mike Barr's experience with teaching membership-based set theory is shared by many mathematicians, and quite a few share his conclusions. One conclusion is that clarification is needed on even more basic questions than just the large/small issue (which concerned Goedel, Mac Lane, and Perez), in order to arrive at conceptions and axiomatizations compatible with the practice of mathematics. For example, I was aiming at such a clarification in pp. 118-128 of my 1976 paper in honor of Professor Eilenberg's 60th birthday, where I advocated some rational connection between conceptualizing and axiomatizing. The complete lack of such a connection in a recent article in the journal "Mathematical Structures in Computer Science" could have been avoided by the editors, if not by the authors. In a section labeled "Basic Set Theory" (p.510) they quote from my above paper a description of the notion of abstract set: 1. ...each element of X has no structure whatsoever. 2. X itself has no internal structure except for equality and inequality of pairs of elements.... immediately followed by their absurd conclusion: "axiomatically this corresponds to taking the membership relation epsilon as the only primitive notion of set theory and to postulating .." some axioms typical to Zermelo-style membership-based theory! Of course those axioms are NOT compatible with the conception quoted: they violate (1) because according to the Zermelo primitives and axioms, an element usually has elements, which would be structure; and they violate (2) since according to those primitives and axioms, a pair of elements of X may stand itself in the membership relation, which would be an internal structure other than equality. The authors neglected to quote the third clause which (as in the example that Mike mentions) their axioms also violate. The notion of abstract set (Kardinalen in Cantor's sense) is basic among the many other notions of cohesive and/or variable sets (Mengen) to the extent that we can model the Mengen via diagrams of maps between abstract sets. Abstract sets may be "abstracted from" less abstract sets, as Cantor did, or used, as most modern mathematics in practice does, as all-purpose memory cells or parameterizers or nodes in such diagrams. In addition to the papers by Colin McLarty mentioned in his message of November 11, 1998, the following papers should help to clarify this notion and its role in mathematics. J. Isbell Adequate sub-categories Illinois J. Math. vol. 4, pp 541-552, 1960 F. W. Lawvere An elementary theory of the category of sets Proc. Nat. Acad. Sc. USA, vol. 52, 1964, pp 1506-1511 F. W. Lawvere Variable quantities and variable structures in topoi, (see especially pp 118-128) in Algebra, Topology, and Category Theory, ed. Heller & Tierney, Academic Press, 1976 F. W. Lawvere Cohesive toposes and Cantor's lauter Einsen (concerning Cantor's neglected Kardinalen) Philosophia Matematica, vol. 2, 1994, pp 5 - 15 W. Mitchell Boolean topoi and the theory of sets (the membership-free content of Goedels constructible sets still needs to be clarified further) Journal of Pure and Applied Algebra, vol. 2, 1972, pp 261-274 ******************************************************************** F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA *********************************************************************