Cantor Confusion
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From: MoeBlee on 23 Jan 2007 12:33 Andy Smith wrote: > If I was asked to sum it up, at present I would say that my > understanding is that you can't have an actually infinite integer, but > reals can be defined as having an actually infinite binary > representation .. (with apologies for the adjective "actually"). So no > surprise that the reals are "uncountable". The uncountability of the reals goes deeper than that. All complete ordered fields are isomorphic with one another. So, irrespective of decimal representations, any way you define the set of real numbers, as long as it provides for a system that is a complete ordered field, the set of real will be uncountable. MoeBlee
From: G. Frege on 23 Jan 2007 12:41 On Tue, 23 Jan 2007 17:15:52 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > > "an infinite set" is an abstract mental concept. > Well, there are different /philosophical/ point of views concerning that question (i.e. the "nature" -the ontology- of sets). On the other hand, I'd prefer a more pragmatic approach here: (infinite) sets, are just _mathematical objects_ we are concerned with in set theory. (At least this is a useful mental picture, imho.) > > If you see this as straightforward it is because your mindset has been > conditioned by your education to see this as normal. > Right. That's what a _mathematical education_ is for. (Won't you think so?) > > I can safely say that if your concepts of infinite sets was placed in > front of the population at large 99% would think that [...] > Who cares?! 99% (or more) are no mathematicians (or at least seriously concerned with set theory). > > Cantor provided a perspective to view infinity, and his insight > underpins, as I understand it, modern set theory. > I guess, that's a reasonable point of view. > > It may be consistent, ... > It most certainly is. > > but I don't see that the philosophical rational is trivial ... > It isn't. You might try to get a copy of Rudy Rucker, Infinity and the mind. The science and philosophy of the infinite. A very nice book. > > ...and, as I understand it, in Cantor's day there were many eminent and > far from stupid mathematicians who couldn't get a handle on it. > There were SOME of them, that's right. A quote from Herb Enderton's Elements of Set Theory: "Cantor's work was well received by some of the prominent mathematicians of his day, such as Richard Dedekind. But his willingness to regard infinite sets as objects to be treated in much the same way as finite sets was bitterly attacked by others, particularly Kronecker. There was no objection to a 'potential infinity' in the form of an unending process, but an 'actual infinity' in the form of a completed infinite set was harder to accept." > > From posts on this site I can see that their descendants are still here > and active ... > ....NONE of which is a professional mathematician. (Does that ring a bell? ;-) > > If I was asked to sum it up, at present I would say that my > understanding is that you can't have an actually infinite integer, but > reals can be defined as having an [...] infinite binary > representation ... So no surprise that the reals are "uncountable". > Well, actually, it WAS a surprise. (Some still aren't able to get it. :-) F. -- E-mail: info<at>simple-line<dot>de
From: stephen on 23 Jan 2007 12:47 Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > stephen(a)nomail.com writes >>Cantor's argument simply says that given a list of real numbers, finite >>or infinite, there exists a real not on the list. That is all. >>An immediate consequence is that there does not exist a list >>that contains all the reals. >> >>> OK, well I do see the argument better now, but if that was an argument >>> that I had suggested for the first time, you would be on me like wolves >>> ... >> >>You still seem to be missing it. > Maybe. The issue wasn't with a finite list, it was whether you could > have an infinite list when all the indices of the rows i.e. all natural > numbers, must be finite ... resolved by considering the list as "an > infinite set" just as the set of "all natural numbers" can be considered > as "an infinite set", even though no member of the natural numbers are > infinite; "an infinite set" is an abstract mental concept. Of course it is an abstract mental concept. So is zero. Mathematics deals with abstractions. > If you see this as straightforward it is because your mindset has been > conditioned by your education to see this as normal. I can safely say > that if your concepts of infinite sets was placed in front of the > population at large 99 % would think that this is barking mad > doublethink ... It is straightforward if you start with the definitions and follow the logic. The fact that many people cannot follow basic logic is not an argument against logic. The fact that many people refuse to stick to actual definitions and instead work with vague undefined terms is not an argument against definitions. You seem to be reading far too much into the word 'infinite'. It has a very simple definition in set theory. There is no point in dragging in unnecessary philosophical baggage. Stephen
From: G. Frege on 23 Jan 2007 13:02 On Tue, 23 Jan 2007 12:52:11 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > There is some perverse pleasure in winning an argument with someone even > if the other person has no clue that you have won. And, the fact that > you are winning is much clearer (at least to you and to intelligent > bystanders) when arguing about math than when arguing about, say, > politics. > Hmmm... hmmm... I think the point (for me) is that *I* am interested in the matters itself we are talking about (here). With other words, when discussing a topic I am aiming for (the) truth, not for winning an argument. (At least this is my "ideal".) >> >> [...] Surely you will have noticed that there is (literally) NO progress >> when arguing with WM. (Imho it's extremely nonsensical to "argue" with >> such a guy.) >> > I basically agree, although whether it is extremely nonsensical depends > on the goal. What do you suggest we do with cranks? Just ignore them? > Basically, yes (I'd say). Torkel Franzen once wrote (in this NG): "[...] Wolfgang M�ckenheim is a classic crank. Why do you imagine, as you seem to do, that there is any point arguing with him?" F. -- E-mail: info<at>simple-line<dot>de
From: David Marcus on 23 Jan 2007 13:07
Dik T. Winter wrote: > In article <1169547646.733664.94830(a)m58g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > > > If the union of singletons {1} U {2} U {3} U... U {n} U ... is an > > > > infinite number omega, then the union of domains of sequences {1} U {1, > > > > 2} U {1, 2, 3} U... U {1, 2, 3, ..., n} U ... is the domain of an > > > > infinite seqeunce {1, 2, 3, ...}. > > > > > > Pray explain what you understand under "domain of", and what you understand > > > under "union of domains". That is complete non-standard terminology. > > > > The domain of a sequence is a natural number. The domain of an infinite > > sequence is omega. > > That is not an explication. That is obfuscation. That is, it is still > clear as mud. Let me see if I can translate. When WM says, "sequence {1}", he means the sequence x with x_1 = 1. When he says, "sequence {1,2}", he means the sequence y with y_1 = 1, y_2 = 2. The "sequence {1,2,3,...}" is the sequence z with z_n = n, for n = 1,2,3,... The domain of x is {1}, which WM says is 1 (since he identifies natural numbers and ordinals, but doesn't like zero). The domain of y is {1,2} = 2. The domain of z is {1,2,3,...} = omega. -- David Marcus |