Cantor Confusion
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From: imaginatorium on 19 Nov 2006 06:43 David Marcus wrote: > imaginatorium(a)despammed.com wrote: > > David Marcus wrote: > > > imaginatorium(a)despammed.com wrote: > > > > Dik T. Winter wrote: > > > > > In article <1163602667.089204.113210(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > > ... > > > > > > > In > > > > > > > principle no axiom is necessary. But you need a few to have some start > > > > > > > to work with. > > > > > > > > > > > > That's the question. By means of axioms you can produce conditional > > > > > > truth at most. I am interested in absolute truth. Axioms will not help > > > > > > us to find it. I don't think we need any axioms. > > > > > > > > > > If you want to find absolute truth you should not look at mathematics. > > > > > > > > Really? There are two groups of order 4; could any truth be more > > > > absolute than that? > > > > > > I think it depends on what the words mean. If the axioms are correct in > > > your model, then the theorems are correct in your model. > > > > Well, model-schmodel, really. (This stuff is a bit beyond me, > > actually...) > > > > It's not entirely clear what the notion of "absolute truth" refers to. > > Suppose you think it is a matter of absolute truth that all men are > > created equal. Then you go to Venus and discover that in their language > > the word 'All' means flying, 'men' means pigs, 'are' means eat, > > 'created' means chocolate, and 'equal' means icecream.* Moreover the > > atmosphere of Venus turns out to be full of flying pigs, but is of such > > chemical composition that icecream of any flavour self-combusts > > explosively. Well, has absolute truth varied? I think the reasonable > > answer is 'No', because a truth is _about_ something, not merely a > > string of formal symbols. > > > > : * Language doesn't work like this - I know, but I haven't time to > > assemble grammars and whatnot > > : just to make the same point. Anyway, see the Hilary Putnam stuff > > about horses and schmorses > > : (which I have only read secondhand in Dennett). > > > > > Why did you pick the statement you did, rather than something like 2 + 2 > > > = 4? > > > > Because as far as I know there is no (normal, sane) interpretation of > > the _words_ of my statement about groups of order 4 other than the > > standard one. Whereas, for example, in other contexts 2 + 2 = 1, so > > while the truth to which "2+2 = 4" refers is absolute, it takes longer > > to write, because you have to spell out the full context, and in > > present crank company even saying "integers" may take 2-3 lines. > > > > You say this depends on my axioms and my model; but are there such that > > make my claim about groups of order 4 untrue? > > I don't know. You claim to have a PhD in mathematics, and you "don't know"? What a feeble answer. So disappointed was I when I saw it, that I was tempted to say I begin to understand where Lester gets his "ideas" from, though mercifully I overcame that temptation, seeing it would probably start him off again. I think I see that there could be a set of rather weak axioms that formed something called groupette theorino, which were simply powerless to prove the existence of two groups of order 4, but my suggestion is that no-one would accept such a miniature as being grownup group theory. Brian Chandler http://imaginatorium.org
From: mueckenh on 19 Nov 2006 07:19 Virgil schrieb: > In article <1163856451.247199.105160(a)h54g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Lester Zick schrieb: > > > > > >Sqrt(2) does exist as the diagonal of the square. But I call that an > > > >idea in order to distinguish it from numbers which can be written in > > > >lists and can be subject to a diagonal proof. I call only those > > > >entities numbers which can be put in trichotomy with each other. > > > > > > I don't know what a diagonal proof and trichotomy may be. > > > > Excuse me. > > 1) The diagonal proof by Cantor shows that any list of real numbers is > > incomplete. For this purpose we use a list (= injective sequence) of > > real numbers and exchange the n-th digit of the n-th number. The > > changed digits put together yield a real number which differs from each > > list entry at least at one position (it differs at position n from the > > n-th list entry). This proof requires that all digits of numbers like > > sqrt(2) do exist and can be exchanged. That assumption is wrong. > > WRONG AGAIN! Cantor's "diagonal" proof merely requires that the nth > number in the list have, in principle even if not in practice, a > determinable nth digit, which is quite a different issue. If not all digits of the diagonal number exist, then the diagonal number is not an example for a number which exists but is not in the list. > > For example, it is definitely the case that for any given position n, > the nth digit of sqrt(2) is, at least in principle, determinable, > however impractical it might be to carry out such a determination. No, it is clear for everyone who is not a fanatic that in principle and in praxis not every digit of sqrt(2) is determinable. Regards, WM
From: Franziska Neugebauer on 19 Nov 2006 07:27 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: [...] >> > But it does not say that it has an ordinal number and a cardinal >> > number. My case iia is in complete agreement with the axiom of >> > infinity. >> > >> >> The axiom of infinity says that the set N exists. Your iia says >> >> "we cannot recognize or treat all of its elements" > > This is no contradiction to the axiom. Define "recognize" and "treat". > Compare the proof that the real numbers can be well-ordered. You have neither given a proof for "we cannot recognize or treat all of its elements" nor defined what "recognize" and "treat" means. You simply invented it. > We cannot construct or define or recognize a well ordering. Irrelevant to your claim. In set theory there are no "equal rights" for things which are different. >> The axiom of infinity does not say anything about ordinal or >> cardinal numbers. However, given that the set N exists and >> the defnition of ordinal and cardinal numbers, it is easy to >> see that if N exists it must have both an ordinal and a cardinal >> number. >> > No. You assume the possibility of a bijection of the set with itself. This _existence_ is "assumed" rightly. The function B := { <n, n> | n e omega } is the desired bijection. > That is not proven from the mere existence of the set It _is_ proven using admissible techniques. > if we cannot recognize or treat all of its elements. This is not even a meaningful objection. "Recognize" and "treat" do not exist within contemporary set theory. What comes next? Physicalisms again? This is out of scope of contemporary set theory. > And even if we could, the axiom of infinity does not prove that > infinite ordinal and cardinal numbers are numbers, i.e., that they > stand in trichotomy with each other. So far we have only the two infinite _ordinals_ omega and omega + 1 The following is valid: 1. omega <_ord omega + 1 and the following is not valid: 2. omega =_ord omega + 1 3. omega >_ ord omega + 1 Hence the so far known infinite ordinals comply with trichotomy. F. N. -- xyz
From: mueckenh on 19 Nov 2006 07:28 Franziska Neugebauer schrieb: > > Please look it up in any modern text book. You will find there that > > omega is a cardinal number too. > > Anyway, omega is not a /natural/ number. And a cow is not a horse. Does that eliminate your error of claiming omega =/= |omega| in modern set theory? > > >> > You need not interpret n as a set (though you can do it). > >> > >> In contemporary set theory almost everything is a set. > > > > In ZFC everything is a set. Not in every set theory. > > I don't want to debate about the axioms of ZFC being sets. The point is > that you want to talk about "columns" which neither ZFC nor any other > contemporary set theory is about. I think that one can introduce new expressions, examples, and illustrations if they have been made sufficiently clear to a correspondent of average IQ. The reaction of William confirms that my ideas are understandable. So your reaction does not concern my writings but rather your means of reception. > > >> A treatise in which variables ("n") and number symbols ("0", "1", > >> ...) do not refer to sets is not a treatise _on_ set theory but > >> a treatise of _application_ of set theory, if ever. > > > > Don't mistake set theory with ZF or ZFC. > > A theory of _sets_ is not a theory of _columns_. Experience has shown that practically all notions used in contemporary mathematics can be defined, and their mathematical properties derived, in ZFC. In this sense, the axiomatic set theory serves as a satisfactory foundation for al other branches of mathematics. It can describe every mathematical notion --- with the exception of what a column is? > >> In contemporary set theory it is said that omega is _the_ _set_ of > >> natural numbers [as Virgil pointed out the _ordered_ set]. The number > >> (cardinality) of omega is named aleph_0. So it is said that the > >> number (cardinality) of _the_ _set_ of natural numbers is aleph_0. > > > > You are completely in error. The number (Anzahl) of a set is its > > ordinal number. > > "Anzahl" is your (or a historical person's) wording not mine. I can't > spot any error in my wording. That is due to your incompetence. > > > Sometimes it is necessary to quote. In particular if you are > > uninformed but nevertheless refuse to take advice from me. > > I don't need any advice from you. You don't know it. That' s why you would need to learn a lot. Look, you have learned from me meanwhile, even against your furious opposition, that omega = |omega| in modern set theory. Doesn't this case make you wonder whether there are other things which you do not yet know but which you could learn from me? > My opinion is > > A (n e omega & |{0, 1, 2, ..., n}| < |omega|) Seems an empty opinion. What is the symbol A refers to? > > >> > As the diagonal is defined to consist of the ends of terms > >> > of lines, this claim is easy to conradict. > >> > >> "ends of terms of lines" is your wording not mine. > > > > It is not your wording, because you prefer to veil your > > inconsistencies, but it is your opinion. > > My opinion is > > |(d_nn) n e omega| = |omega| I know. You say there are less natural numbers than omega, but there are as many natural numbers as omega. > > > Can you understand mathematical symbols? Can you understand d_nn <--> > > n? > > Writing down a bunch of symbols does not mean that you created a > mathematical notation. Rephrase what precisely "There are more natural > numbers d_nn than natural numbers n." shall mean. It means that we have discovered an inconsistency in the assumption that there were infinitely many finite numbers. Regards, WM
From: mueckenh on 19 Nov 2006 07:33
Franziska Neugebauer schrieb: > >> > The cardinality of omega is omega. > >> > >> The cardinality of omega is |omega| not omega. > > > > Kunen's "Set Theory" defines |A| to be the least ordinal that can be > > bijected with A. So, with this definition, |omega| = omega. > > You are absolutely right. Well learned (after all)! Now try to understand the next step: If omega exists, then |omega| =/= omega & |omega| = omega. Then you will have reached a higher level of understanding math than most mathematicians. Regards, WM |