Cantor Confusion
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From: William Hughes on 19 Nov 2006 08:31 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > > > > > > > > > > 1 > > > > > > > 2 > > > > > > > 3 > > > > > > > ... > > > > > > > n <--> 1,2,3,...n > > > > > > > > > > > > Please distinguish: > > > > > iia: There is no number counting the elements of N. > > > > > iib: There is a number omega counting the elements of N. > > > > > > > No. You are trying to show that assuming case iia leads > > to a contradiction. > > No. Case iia does not lead to a contradiction. > > > To do this you need to assume case iia. In > > particular > > you are assuming > > > > - the set, N, of all natural numbers exists > > - the set N is infinite. > > - N has no last element > > Yes. > > > > > The diagonal contains all the d_nn. No line contains all the d_nn. > > Therefore the diagonal is longer than every line. > > The diagonal consists of line indexes, i.e., of the line ends. > Therefore it is a subset of the line indexes. > Yes, the diagonal is the set of line indexes (a subset, but not a proper subset). We need more than this. Consider the set X={1,2} and the two finite sequences A and B A= {1,2,1,2} B={1,2} A and B both consist of elements of the set X, but A is longer than B. So to say that both the diagonal and the lines are subsets of the line indexes is not enough to show that A cannot be longer than B. So you must have something more in mind when you claim that the fact that every element of the diagonal is a line index means that the diagonal cannot be longer than every line. What is this? Recall: every line is shorter than the diagonal; there is no longest line. - William Hughes
From: Franziska Neugebauer on 19 Nov 2006 08:32 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> Virgil wrote: >> >> > In article <1163856355.707119.306700(a)m7g2000cwm.googlegroups.com>, >> > mueckenh(a)rz.fh-augsburg.de wrote: >> > >> >> Virgil schrieb: >> [...] >> >> And there are lines as long as the diagonal is. >> > >> > Name one. > > The elements of the diagonal are a subset of the line ends. Though Virgil posed this question: Name one single _line_ which is as long as the diagonal ist. F. N. -- xyz
From: Franziska Neugebauer on 19 Nov 2006 09:23 ADDENDUM mueckenh(a)rz.fh-augsburg.de wrote: > 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 got me wrong and this is obviously _my_ fault. I wrote ,----[ <455f1091$0$97214$892e7fe2(a)authen.yellow.readfreenews.net> ] | The cardinality of omega is |omega| not omega. `---- This may mistakenly be interpreted as the proposition |omega| =/= omega (*) which I did not want to posit. In general "cardinality of set X" is usually written "card(X)" or "| X |" and not "X". In the case of the sets 0, 1, ..., omega | X | does equal X under the common cardinal assignment as David Marcus has already pointed out: ,----[ <MPG.1fc9337dea93ba82989932(a)news.rcn.com> ] | Kunen's "Set Theory" defines |A| to be the least ordinal that can be | bijected with A. So, with this definition, |omega| = omega. `---- F. N. -- xyz
From: David Marcus on 19 Nov 2006 11:16 imaginatorium(a)despammed.com wrote: > > 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. Are you making a joke? > 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. Obviously, if we throw out all the axioms, we can't do anything. But, if that was your question, you could answer it yourself. I don't know if there is a more interesting answer. I was also thinking that it depends on what you mean by "absolutely true". This seems to relate to your statement that "no one would accept" such a theory. It is hard to say what people will accept. -- David Marcus
From: David Marcus on 19 Nov 2006 11:23
Franziska Neugebauer wrote: > I do not speculate about what is describable and what not. If you want > to posit a definition do so! > > It is curious that you obviously don't like to give precise definitions > of certain notions even when you have explicitly been asked for. You've only been here a short time, so you probably missed my discussion with WM about what the word "definition" means. His definition of the word "definition" is very different from ours. To him, a "definition" is any sort of discussion/explanation. He denies that our sort of definition is possible or reasonable. -- David Marcus |