Gerry Myerson wrote: > 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.
Likewise, if I accept CH, I have no problem "seeing" what you /mean/ by "a set having cardinality strictly between that of the integers and that of the reals". However, I won't accept that such a thing is actually a set.
> 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.
And in our CH example, I can hold (philosophically) that the situation is similar: we can agree to talk about objects that exist in ZFC which don't exist in ZFC+CH, discuss proofs of them etc. I will simply ask that, to reduce confusion, you don't call such objects "sets", but rather call them "weak sets" or "pre-sets", or some other precious and denigrating thing ;)
> > > 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. >
This is the assertion that I don't know if I agree with or not. What do we really mean by "exists"? Is it a statement about the outcome of applying a given set of rules, or does is it a more "concrete" statement about whether something is "real" or "imagined"?
Obviously if I assume CH, the given theorem is false. But /why/ do I assume CH to start with? You seem to be saying that the only reason I would assert this would be if I thought such a set "really" doesn't exist in some Platonic sense.
But if I look at the example of AoC, I feel less secure in this. I am happy to make use of the axiom, because, to be honest, without it it is much harder (at times, even impossible) to prove certain things. But I'm also happy to consider the universe provided by ZF alone, because it is also beautiful. So which universe am I claiming is the "real" universe of "sets"? (oh, and since I'm assuming you can read my mind, where /did/ I put my car keys?)
> 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. >
I never claimed I was either original or profound :).
I have to add that one of the things that is nice about axiomatic systems is that there is no need to take a philosophical stance on these issues in order to determine if a particular proof holds in ZFC.