"David C. Ullrich" wrote: > > On Wed, 25 Jul 2001 23:19:48 +0100, Peter Percival > <firstname.lastname@example.org> wrote: > > >"David C. Ullrich" wrote: > >> > >> ... > >> > >> I'm almost certain I'm confused about this: In second-order > >> logic we quantify over sets, right? Now it seems like the sets > >> we quantify over must be "real sets", not just the sets in some > >> model of set theory. But for those of us who don't believe > >> that it makes any sense to talk about those real sets this is > >> a problem... > > > >I appreciate that this is going OT but may I ask if you disagree with a > >statement I made elsewhere: a real set is an element of the cumulative > >hierarchy, with a problem about where to begin (i.e. what urelements, if > >any) and a problem about how far to go (i.e. what axiom of infinity to > >adopt). > > If I honestly don't see what sense something makes I can't see why > you would ask me questions about it. I'll give a better answer > after I see your reply to this question: Do you agree that freebles > are actually fredly?
I am unable to form an opinion, but if the words freebles and fredly were used in ordinary English the way that set is I might have a rough idea.
> > Seriously. First, there's _really_ no reason anyone should care > about _my_ opinion on this. That cumulative heirachy thing
Why not? It's no less interesting than the opinion of others.
> makes much more sense to me than it did years ago when I was > even stupider than at present. But I honestly don't know > exactly what people mean when they talk about real sets.
Fine. "Shirt" is an ordinary-enough word but do you know _exactly_ what someone means if they say that they have to sew a button on a shirt? With all the possibilities of fabric, cut and colour, no you don't. Nevertheless you could hold a conversation in which such a thing was said. Do you know inexactly what people mean when they talk about real sets? If so, there is some similarity with shirts.
> > >I realize that even if you agree with it you might not find it a > >_useful_ answer to the question: what is a real set? > > With the same disclaimers, I don't see how it is a useful question > to _ask_, much less try to answer. If we _do_ somehow agree on > what a real set is there will be a definition that we have > agreed on. Will we agree on the definition of all the terms in > that definition? No, there will be at least one undefined term. > I don't see any reason not to just take "set" as undefined.
Nor me. Note that to learn ordinary language we have to understand words that are not defined. At least they are not defined for us when we learn how to use them competently. The fact that we could look them up in a dictionary later does not invalidate what I'm saying.
> > > because of the > >doubts at the two "ends" so to speak. But still it would be a start. > >On the other hand you may disagree more radically. Perhaps you have > >worries about the way the cumulative hierarchy is built: e.g. if X is an > >infinite set, then we know very little about P(X). I.e. it's not just > >the ends that are a problem, it's much of what is between them as well. > > > >When you write "don't believe that it makes any sense to talk about > >those real sets" do you mean that there may be real sets but one can say > >nothing mathematically sensible about them. E.g. this might apply to > >"set of tools in my toolbox". Or do you believe that there are no such > >things as real sets to talk about? > > Neither. To say either of those things I'd have to first know > what we're talking about. > > Seriously this time: Do you believe that freebles are real? > Probably not. So does that mean you think there are freebles > but we can't say anything about them? Or do you believe that > you know what a freeble is and that there simply is no such > thing? Neither: You don't know the meaning of the word > "freeble", and trying to answer my questions about freebles > given that you don't know what the word means strikes you > as silly, right?
I will never have any idea what freebles are because I will never come across the word outside this discussion. _If_ I did come across it elsewhere I might begin to acquire an inkling of what it meant--not a definition but an informal idea. I might get just such an idea what set meant. Since you invented freeble to have absolutely no meaning for anyone (right?) the analogy with set is imprecise. I'll not hold that against you since all analogies are imprecise.
> > However we decide this there _will_ be at least one word > the meaning of which we have _not_ settled, nor even attempted > to discuss precisely. It's up to us to decide which words
But you would allow imprecise discussion? The claim that a set is something in the cumulative hierarchy might be part of the imprecise discussion. Imprecise because one doesn't know precisely where the cumulative hierarchy begins or ends or how precisely one gets from one level to the next.
> will be in this class. Putting "set" (and more important > "is an element of") into this class strikes me as an > excellent choice.
Me to. Now cook up some axioms for set and is an element of. Are you guided by a, perhaps rather rough and ready, idea of what set and is an element of mean? If not, how do you choose your axioms? I don't doubt that you might choose some axioms and change them later after studying them, but you will have to start somewhere.
> > > Someone who countered: what about > >"the set of tools in my tool box"? might get the reply: that is an > >unnecessary linguistic variant of "the tools in my tool box". > > > >Cheers, PP > > > > David C. Ullrich