Reinhard.Boerger@FernUni-Hagen.de wrote:
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?
cardinal< aleph_0. This is useful notationally as omega*2 refers to ordinal multiplication (so it has the same order type as omega \times 2 with lexicographic order) and should not confused with cardinal multiplication aleph_0 * 2, which of course is just aleph_0. I mention
I have always wondered why when talking about toposes from sets, people only mention "all sets of cardinality < something". Unless you have classes of some kind (with the attendant problem of making sense of categories of categories), it doesn't even make sense. If a is any limit ordinal, the category of all sets of rank < a is a topos. [for those who might have forgotten: Define V_a for >ordinals< a by transfinite induction as V_0 is the empty set, V_{a+1} is the power set of V_a, if a is a limit ordinal V_a is the union of V_b for b < a. Rank is very different from cardinality: {V_a} has rank a+1, which can be very large, but has cardinality 1, as small as it can get for any non-empty set.] [Another parenthetical remarks may be in order: I am making the customary intentional distinction between the >ordinal< omega and the this explicitly because, once, I saw V_{omega*2} taken to be as "the 'set' of all sets of cardinality < omega*2", whatever that may mean. Also, recently I noticed that in "Sketches of an Elephant", the first infinite cardinal referred to as $\omega$.] If a is a limit ordinal, but not the first, then V_a has a natural number object. So, in some sense V_{omega*2} is (very) small boolean topos with a natural number object. I find it useful to keep this example in mind when thinking about boolean toposes. For example, I would be very surprised if anything like Adams completion or Bousefield completion exist V_{omega*2}. [Bousfield completion can be made functorial before passing to the homotopy category. This is an useful fact. But the original aim can be expressed without functors: For any space X, there is a map X \to Y such that ...] So, I find the claim that mathematics that does not refer to 'all sets' etc can be done in toposes. Switching back to the original topic: Given ZF(C) - replacement, Fraenkel's version of replacement will prove the same theorems (just about sets) as reflection: To ZF(C) - replacement, add the assumption of an internal universe V that is elementarily equivalent to the 'whole universe'. It would be interesting to look at a topos theoretic version, but I have no idea of what the language of an internal topos means, much less how to say such things as "the natural number object of the internal topos T is the natural number object of the containing topos' [perhaps it is "N \times T \to T" is a small fibration, and the corresponding internal category of T is the natural number of object of T", but what is an internal category of an internal category?] Note that in the above, V will be an V_a, but a need not be a cardinal; for all I know, it may be consistent for a to have cofinality omega as an ordinal of the "big universe". Thus inaccessibility is a red herring when talking about a universe of sets that satisfy ZFC. BTW, do we know if Fraenkels' version of replacement (replacement for predicative functions as opposed to replacement for arbitrary functions) is inadequate for theorems "in the wild" that talk only about sets? [So theorems about "the category of all ...", or, say, model categories where the factorizations need not be given by predicative functions, do not count.] In other words, does mathematics really need Grothendieck universes? Nath Rao 16-Mar-2005 11:19:02 -0400,7295;000000000000-00000000