Cantor Confusion
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From: mueckenh on 21 Nov 2006 11:26 Dik T. Winter schrieb: > > If you assert that the diagonal can be longer than any line, then you > > have no reasons, because the diagonal consists of line elements and > > cannot be where no line is. So your second assertion is outside of > > logic and outside of any mathematics. Therefore I am not willing to > > discuss this topic further. > > You do not want to discuss it because you are not able to prove it. The set D of indexes of the diagonal is a subset of the sets L_n of indexes of the lines.: D - UL_n = empty set. > But > indeed, let's stop this non-discussion. Unless you come with a proof. > You are not even able to comprehend potential infinity. Tell me which point is not accepted: 1) Every line which contains an index of the diagonal, contains all preceding indexes of the diagonal too. 2) Every index of the diagonal is in a line. 3) In order to show that there is no line containing all indexes of the diagonal, there must be found at least one index, which is in the diagonal but not in any line. This is impossible. 4) There is no line with infinitely many indexes. 5) Conclusion: There is no diagonal with infinitely many indexes. Regards, WM
From: mueckenh on 21 Nov 2006 11:29 Dik T. Winter schrieb: > > > Well, the same in set theory. In one form it has AC as a fundamental > > > truth, in another version it is not a fundamental truth. What is the > > > essential difference between the cases? > > > > > That there is not an "underlying plane" which it is related to. > > So there should be an "underlying plane", whatever that may mean? For geometry, it is required, yes, for arithmetic it is not. > What > is the "underlying plane" of hyperbolic geometry? What is it in elliptic > geometry? The corresponding spaces and surfaces. > > The reason is that these "numbers" have no decimal representation, > > hence cannot be used to prove that they are uncountably many. > > So what was your vehemence directed at? Not at calling them numbers, > because you do so yourself. That they have no decimal representation? > But that was not in discussion at that time. So I wonder. > > > > What is the essential difference between all those cases? I once asked > > > you for a definition of "number", and you did never supply a proper > > > definition. Now you say yourself that the definition has been changed in > > > the course of time. I may note that there is no trichonomy between the > > > complex numbers, it is not possible to consistently order them. > > > > You may call these entities numbers or not. My opposition stems from > > their use in Cantor's list and the lacking trichotomy. > > So, now we may call them numbers (as you do). And, we are back to basics. > Your misunderstanding of Cantor's argument and whatever. They have no representation which could be used for any form of Cantor's list. Regards, WM
From: mueckenh on 21 Nov 2006 11:31 Dik T. Winter schrieb: > > > There are infinite paths > > > in your tree, but they do not contain a node that represents (for > > > instance) 1/3. So, if the nodes represent numbers (as you have said), > > > > Do you have a reference? > > Not needed. Just above you state that the nodes represent the bits 0 or > 1. I have shown how you could concatenate the representation of a node > with the representations of its parent nodes to get a number. A number consists of bits. Some numbers consist even of one bit. But you must not mix up these terms. A number like 1/2 consist of the bit sequence 0.1000.... That is a path in my tree: 0. | 1 | 0 | 0 .... > > > > 1/3 is not in your tree. Of course it is, like 0.333... is in Cantor's list of decimals. > You are not clear about what the numbers in > > > your tree are. Are they the nodes? Are they the paths? Sometimes > > > you say one thing other times you say something different. So to get > > > proper understanding. What are the things that represent numbers? > > > > Infinite paths. > > I do not understand. You stated the nodes represent bits. Yes. That's the stuff numbers are built from. Regards, WM
From: mueckenh on 21 Nov 2006 11:34 Randy Poe schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Randy Poe schrieb: > > > > > > > > I know that at most 10^100 digits of sqrt(2) can be determined, in > > > > principle. > > > > > > In principle, if a is the sqrt(2) to 10^100 digits, then > > > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. > > > > > > What "in principle" prevents me from calculating 2/a, > > > or adding it to 1, or taking 0.5*(a + 2/a)? > > > > Lack of bits to represent 2/a if a already requires all bits available. > > That means it's hard in practice. > > But what prevents 2/a from existing in principle? It is not "hard in practice", but it is impossible. There is no chance. That means "in principle". > > > If you don't believe me, simply try it. By about 330 calculations you > > should be able to produce the first 10^100 digits. If you have a slow > > computer you will need one second per calculation. So let it run for 5 > > minutes and you will know what prevents you from calculating 2/a. > > What difference does the speed of an actual computer make? We > aren't talking about implementation, we're talking about "in > principle". > > How can a represent "all bits available" IN PRINCIPLE? In principle, > there is no limit on the available bits. Do you think, in principle there are hidden variables, we only can't determine them? No, in principle we cannot circumvent the uncertainty relations and we cannot do what we know to be excluded in eternity. That is a meaningful understanding of "in principle". If we accept your "in principle", then in principle we can maintain the most ridiculous mess - and then we arrive at the finished infinity. Regards, WM
From: mueckenh on 21 Nov 2006 11:38
David R Tribble schrieb: > david petry schrieb: > >> Perhaps we should replace "absolute truth" with "culturally neutral > >> truth", or in other words, truth without any cultural, religious, or > >> philosophical bias. [...] Thinking about this question > >> leads most of us to believe that there is a core of mathematics which > >> every such civilization will accept. > > > > mueckenh wrote: > >> without axioms, yes. For instance: I + I = II (after translating "+" > >> and "="). Therefore I call this an absolute truth. > > > > David R Tribble schrieb: > >> Which axioms are you using to describe the "+" and "=" operators? > > > > mueckenh wrote: > > Axioms? For which purpose? > > For the purpose of making sense of the phrase "I + I = II". > > > > Do you think the symbols constituting the > > words constituting the axioms are easier or clearer to understand than > > the symbols "+" and "="? > > Without words that define those symbols, the symbols are > meaningless. I can say that "x =& y", but without any definition > of "x", "=&", and "y" it is just a meaningless string of symbols. Without symbols and actions that define those words, the words are meaninless. > > > > Take an apple and then another apple. Show > > the apples first apart and then together. Repeat with oranges or > > fingers or mixed objects, possibly. That defines all that is needed. > > So then we can take yours words there as axiomatic definitions of > "+" and "="? Or are you still claiming that "I + I = II" means > something without axiomatic definitions? What do you mean with "There" or "exists" or "a"? How do you define "set" and "which" and "has"? What is the meaning of "no" and "elements"? I think these symbols and expressions (like "exists" and in particula, " there") deserve closer examination than "+" and "II". Regards, WM |