In article <firstname.lastname@example.org>, email@example.com wrote:
> Gerry Myerson wrote: > > In article <firstname.lastname@example.org>, > > email@example.com wrote: > > > > > Let us simply restrict ourselves to the cases where one makes no > > > assumption regarding CH, and where one asserts CH. > > > > > > How is this conceptually different for you than considering on the one > > > hand the usual axioms of group theory, and on the other assuming in > > > addition commutativity, so that all groups are Abelian? > > > > For me, it's very different, as follows. > > If I change the definition of a group to include commutativity, > > S_3 still exists; it's not a group any more, but it is > > a mathematical object about which I can prove theorems. > > If I change the definition, and you don't, you and I still have > > the same universe of mathematical objects - we just don't use > > the same words to describe them. > > > > > But if you accept CH, and I don't, then I have sets of real numbers > > that are intermediate in cardinality between the integers and the reals, > > and you don't. I can prove theorems about these things, and you > > can't even see them. That (to me) is a big conceptual difference > > between assuming commutativity and assuming CH. > > > > I preface my remarks by admitting that my knowledge of the topics of > logic and set theory amounts largely to "what I've been able to pick up > on the internet". Hopefully, I will not make a complete fool of myself. > > First, if I accept the axiom of commutativity ("AoCm"), and you don't, > then we will both have simple finite groups; but you will have objects > (simple finite groups of non-prime order) which I don't. "There exists > a simple group of order 60" will be a theorem for you, and not for me; > since I can't even "see" A_5 as an object under consideration.
You won't have any trouble seeing A_5. You just won't accept that it is a group. You and I will be able to talk about it, and prove theorems about it, provided I agree to call it an almost-group or a not-quite-group or something like that, and don't insist on calling it a group. You and I will have the same objects, just different terminology.
> That seems no different from the state of affairs when I assume CH and > you do not. For example, I will consider any theorem of the form "If A > is such that w < card(A) < 2^w, then (whatever)" to follow vacuously in > my system as easily as the theorem "if ab != ba, then (whatever)" > follows for Abelian groups.
But what will you make of the theorem, "There exists a set A of reals with cardinality strictly between that of the integers and that of the reals"? This isn't a question of terminology - you and I disagree about the existence of this set, not about what to call it.
I'll snip the well-foundedness stuff, as I have nought to say on't.
> One problem (at least!) that I can see with my analogy is that group > theory makes no claims to being foundational - one simply /must/ admit > that there are things that are perfectly sensible to consider to be > "mathematical", which are not groups, Abelian or otherwise. > > Set theory, on the other hand, seems to be an attempt to assert that if > you are able to say something of "general mathematical interest" in > /any/ language, then you should also be able say it in ZFC (or NBG > or...).
Agreed. I think I said something similar somewhere in this thread.
-- Gerry Myerson (firstname.lastname@example.org) (i -> u for email)