Dear colleagues, Reinhard Boerger has appropriately raised (as others before) the question of the relation of "replacement" with categories of sets. As stated in my 1964 - 1965 papers and as understood already before 1920 (1) closure under exponentiation is sufficient for a foundation for mathematics if by mathematics one means algebraic geometry, functional analysis, dynamical systems, partial differential equations, combinatorics, etc.; (2) but closure under exponentiation is not sufficient for certain speculations in which set theorists had developed an interest; if one wished, one could introduce stronger axioms. From the standpoint of a theory in the Frege-Peano style, it might appear that the replacement schema is the principal means for justifying those speculations. However, the left adjoint (Ua incl b iff a incl Pb) to the power set operation P is equally important, exploiting as it does the hidden structure as a FAMILY of sets that any set of such a theory has. The usual "image" version of the replacement operation does not give directly any sets of cardinality larger than its inputs; only in the case where the set it produces is "really" a family (of increasingly large sets, even if few) the union axiom can then give a larger result. On the other hand (assuming the axiom of choice) it is through producing larger cardinals that stronger axioms can express their strength. Thus, for a principle combining the union and replacement principles in a context where families are explicitly recognized as such, I proposed: If E ---> B is such that B exists and the fibers E sub b exist for each b, then E exists. Note that the usual way in geometry (hence in topos theory) for expressing a family indexed by B is by such a fibration over B. (In case the sets in the family are given as subsets of a set X, their union is a quotient of E modulo the usual "nerve" resolution.) Most families that arise in mathematics are easily put into this form, i.e. are not abstractly imposed from outside and hence do not need extra axioms to insure their existence; therefore the "internal limits and colimits" in topos theory can work well as tools for geometrical construction: they are honest adjoints but applied only to "internal families" and universal among internal families. The above existence principle is intended to have several interpretations, depending on how many and what kind of families are to be representable as fibers in the geometrical way. Besides the tautological "internal" answer (sufficient for most purposes) there are two distinct kinds of answers to the question: "From where do we take the families E to which we want to apply the "existence" assertion?" The subjective answer, attributed to Fraenkel and followed by most books on set theory is that we take E from the world of formulas. (For category theory we need to consider formulas of category theory (not epsilon theory), having one free variable ranging over points of B and another ranging over objects of the category and enjoying the "function" hypothesis up to isomorphism.) The other (objective) answer, similar in spirit to the (finitely axiomatizable) Bernays-Goedel theory, is that E lives initially in another category into which the one we are describing is embedded (either as a subcategory or as an internal category object). Such an embedding may or may not be full; when it is, then our objective "replacement" principle is revealed as equivalent to inaccessibility (of the category). As explained in the Appendix of the book "Sets for Mathematics" the demand for more sets has its source in the need to objectively parameterize mathematical objects. I can only recall two occasions where topos theorists needed more parameterizers than those obviously guaranteed by exponentiation and boundedness of geometric morphisms; in such situations one can explicitly assume more about the toposes under investigation (guided by available experience in avoiding inconsistencies). The best context for discussing these matters is the category of categories. However, assumptions of large cardinal strength can be applied equally to a category of categories and to the corresponding topos of discrete categories, since categories can be represented as finite diagrams of discrete categories. Greetings, Bill Lawvere REFERENCES: Elementary Theory of the Category of Sets, Proceedings of the National Academy of Science 52, No. 6 (December 1964), 1506-1511. An Elementary Theory of the Category of Sets, Preprint, University of Chicago, 1965, 32 pages. The Category of Categories as a Foundation for Mathematics, La Jolla Conference on Categorical Algebra, Springer Verlag (1966), 1 - 20. "Sets for Mathematics" with Robert Rosebrugh Cambridge University Press, 2003. On Sat, 12 Mar 2005 Reinhard.Boerger@FernUni-Hagen.de wrote:
Hello,
Carl Futia wrote:
Is it known just how much mathematics can be done using Lawvere's axiom scheme? My confusion on this point arises partly from a remark by Gray in his paper "The categorical comprehension scheme" in which he asserted that Lawvere's axioms couldn't do all that was claimed for them. Of course all of this predates the recognition that an elementary topos (with nn object) can be treated as a "universe of sets".
As I see it, the problem is with the replacement scheme. Even if you start with ZF and classical logic, you have problems to express replacement in terms of first-order properties of the category of sets. An infinite cardinal a is called a strong limit cardinal, if b<a implies 2^a<b. For every strong limit cardinal a the category of all sets of cardinality < a is en elementary topos, regardless whether a is regular or not; regular strong lomit cardinals are inaccessible cardinals. Replacement should yield regularity of a, but how to express it? The easiest way to find a singular strong limit cardinal a is to start with an arbitrary infinite cardinal b(0), define b(n+1):=2^b(n), and a the sum of al b(n) for all natural number n. Then a is obviously singular because it is uncountable but the sum of countably many strictly smaller cardinals; a is a strong limit cardial because every cardinal <a is <b(n) for some n.
Greetings Reinhard
17-Mar-2005 17:13:39 -0400,2360;000000000000-00000000