Cantor Confusion
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From: Eckard Blumschein on 22 Nov 2006 11:33 On 11/21/2006 10:25 PM, Virgil wrote: > In article <4563067B.5050805(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 10/29/2006 10:22 PM, mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Every set of natural numbers has a superset of natural numbers which is >> > finite. Every! >> >> >> I am only aware of the unique natural numbers. If one imagines them >> altogether like a set, then this bag may also be the only lonly one. > > Is that "lonely"? I meant: There is only one natural sequence of numbers: 1, 2, 3, ... For instance 0, 1, 2, ... or 3, 4, 5, ... or 2, 4, 6, ... are something else. "Every" implies that there are more than one.
From: Franziska Neugebauer on 22 Nov 2006 11:45 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Virgil schrieb: >> >> In article <456304B0.70705(a)et.uni-magdeburg.de>, >> >> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > >> >> > Being admittedly not very familiar with set theory, I >> >> > nonetheless wonder if sets are considered like something going >> >> > on forever. >> >> >> >> Sequences do, sets do not. >> >> >> > Sequences are sets. >> >> Is the "identical sequence" n |-> n a set, too? > > In ZF and ZFC everything is a set. Can you give us a representation of the identical sequence in terms of sets? F. N. -- xyz
From: Eckard Blumschein on 22 Nov 2006 12:00 On 11/22/2006 12:19 AM, David Marcus wrote: > Eckard Blumschein wrote: >> On 10/29/2006 10:22 PM, mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Every set of natural numbers has a superset of natural numbers which is >> > finite. Every! >> >> I am only aware of the unique natural numbers. If one imagines them >> altogether like a set, then this bag may also be the only lonly one. > > WM means that every finite set of natural numbers is contained in a > larger finite set of natural numbers. The sequence of natural numbers has no reason to end, exept if one does not have enough power of imagination or abstraction, respectively, to exclude any physical restriction from the ideal model of natural numbers. > Since he believes all sets of > natural numbers are finite, he then concludes that every set of natural > numbers is contained in a finite set of natural numbers. This seems to > be due to an allergy to the word "set". Thank you for clarification. The word set does indeed serve as a deliberately obscuring crutch. It suggests an aprioric point of view, contrasting to Archimedes. WM denies this selfcontradictory abstraction. Following Leibniz, I consider uncountable numbers a useful fiction with a fundamentum in re.
From: Eckard Blumschein on 22 Nov 2006 12:22 On 11/10/2006 3:03 AM, Dik T. Winter wrote: > But in mathematics comparing numbers comes in quite late in > the process of definition. With mathematics you perhaps mean the possibly questionable attempt to formally justify what has proven successful. > It starts when the ordering axioms are > given. In a set a ardering relation may be can be defined. Let us use > >= as the basis. Mathematicians like you are hopefully aware of the trifle that the relation >= cannot be applied to the really real numbers. Do not worry, the actual infinity is not taken seriously with ZF axioms. My problem is: Narrow-minded former learner of set theory cannot imagine the basic idea by Dedekind and Cantor wrong, the whole cardinal ordinal stuff reduced to a simple distinction between finite, countably infinite, and uncountable. The particular obstacle is: really real numbers are uncountable fictions and therefore they exhibit properties pretty different from those of genuine, i.e. rational, numbers.
From: MoeBlee on 22 Nov 2006 13:15
mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > The countability of the set W of all the words of a finite alphabet is > > > proved by other means than the usual countability proof. The latter > > > consists in constructing a bijection with N, i.e., in constructing a > > > list. But it is impossible to construct a list of all words. It is > > > impossible to construct a bijection between W and N. > > > > WRONG. It is a theorem that from the set of finite sequences of members > > of a finite alphabet there is an injection into N. > > Nevertheless it is impossible to construct such an injection WRONG. > as it s > impossible to construct a well order of the reals). Showing an injection from the set of finite sequences of members of a finite alphabet into N is not analogous to well ordering the reals. The former can be explicit; while the later cannot be. You should learn some set theory if you're going to be popping your mouth off about set theory at the rate of dozens of posts a day. > > > Why should it be > > > possible to construct a bijection between N and N? > > > > Because ANY x is 1-1 with x, by the identity function on x. > > It is an unfounded assumption to believe that all x of the set appear > in this bijection, only by writing x = x. Ridiculous! We don't prove the existence of an identity function just by writing 'x = x'. What is ridiculous is your thinking that set theory proposes any such proof. What is ridiculous is your ignorance of set theory while you so fervently shoot your mouth off about it. > > > > > Note however that in order to attach a definite cardinal number to > > > > > every set, well-ordering must be possible for every set. That is why > > > > > Cantor insisted on well-ordering of all sets. > > > > > > > > Piffle. Well-ordering has nothing to do with the existence or not of > > > > at least one bijection. > > But it has to do with the existence of a bijection to an ordinal > number. > > > > > > > You are wrong. If a set cannot be well-ordered, it cannot be assigned a > > > definite cardinal number. In order to assign a cardinal, at least one > > > bijection to an ordinal is required. This requires at least one > > > well-ordering. > > > > WRONG. By using Scott's method (which requires the axiom of > > reqularity), we don't need the well ordering theorem to prove that > > every set has a cardinality. > > No. But we need it to determine this cardinality. Again you think > erroneously that an existence proof is of any value. Constructivity is a separate matter. My point stands that we do not need the well ordering theorem for every set to have a cardinality. > Regards, WM > > PS: What about the binary tree in ZFC? Any results? Any idea why it > can't be done? I already mentioned what I would require to discuss your tree argument. If you want me to talk about your tree, then please answer my previous posts from about a week ago on the subject of even discussing your argument. I put it to you very clearly just what you need to tell me before we proceed. If you won't do so, then I'm not inclined to waste my time on your argument. MoeBlee |