Cantor Confusion
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From: David R Tribble on 6 Dec 2006 20:01 stephen wrote: >> The idea that 25 is ever going to be anything but 25 is absolutely ridiculous. >> The idea that a set ever changes is equally ridiculous. > Brian Chandler wrote: > Obviously not a FORTRAN programmer... > > SUBROUTINE CHANGE(A, B) > IF(A .EQ. 25) A = B > RETURN > END > > Now try: > CALL CHANGE(25, 17) :-) But I still don't see how calling a subroutine changes the value of 25. Unless you have a memory overwrite bug. A better programming metaphor is that numbers and sets are constants, not variables.
From: Dik T. Winter on 6 Dec 2006 20:04 In article <1165369900.708031.124650(a)n67g2000cwd.googlegroups.com> cbrown(a)cbrownsystems.com writes: > Dik T. Winter wrote: .... > > Sorry, I can't read Word documents (I have no reader available). > > But again who requires one-point compactification with what goal? > > His paper "Adaptation of Spectral Analysis to Reality" doesn't contain > the word "compactification" (or even the fragment "compact"). It seems > to promote the use of only the real parts of a Fourier Transform > (because complex values have no "physical reality"; even negative > values are suspect at best) and restricting the time domain to be (-oo, > 0] (because we only know the past and cannot claim to know the future). You mean he considers negative numbers as suspect but does allow only negative numbers for the time domain? And apparently he has never heard about shifting the origin. According to that reasoning we are forever living in the year 0, while the coordinates of all past years go up by one on the first of January every year. And by the same reasoning we can not even state that on the first of January 2007 Slovenia will introduce the euro, because we can not look in the future. Bizarre. > From the paper: > > "7 Mathematical peculiarities > > In order to benefit from complete removal of redundancy, one has to > restrict the field of all real numbers R to the field R+ of positive > ones. Mathematics largely neglected R+ so far. Important operations > like convolution are nonetheless known to be valid within R+, too. What > was the function of time in R, travels outward relative to the zero of > elapsed time. It is permanently incremented at zero. Integration and > its reverse do no add or remove a constant of integration. > Backward-ramp, down-step and singular point constitute a new unified > set of singularity functions with belonging elementary spectra. " Yes, bizarre. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 6 Dec 2006 20:09 In article <1165403079.475990.150660(a)f1g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > > I have difficulty with the tree because your explanations are confused > > > > and sometimes contradictionary. > > > > > > Which one? > > > > Many. > > Care to name one? I disremember, and it is not easy to find among the many articles by you. > Please do not mistake your misunderstandings for > errors of mine. For instance I never stated that nodes represent > numbers, as you erroneously believed. Pray reread what I wrote: "the nodes can be made to represent numbers in your tree". That is an easy exercise, I even did show it. The same for the edges, I did show that too. So actually the nodes and edges also represent numbers in some way. > > > > You state that you are using limits with your infinite paths? > > > > > > Of course. The paths are nothing else but another way of denoting a > > > real number in binary representation. > > > > But in that case you are doing something non-standard. > > Not at all! I represent numbers by standard binary notations. It is using the limits where you are doing something non-standard. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 6 Dec 2006 20:11 In article <1165403177.893469.76590(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > To have a big mouth is not enough to create a world, not even a notion. > > > You would see that if you tried to say where the assumed object > > > existed. > > > > In my mind. I can reasonably think about the set of all natural numbers. > > Honest, I have no problem with it. > > You believe you could. That is a difference to "can". Other people > think they can reasonably think of having an immortal soul. How do you know I can not? How do you know what I can think? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 6 Dec 2006 20:30
In article <1165403362.548786.220370(a)16g2000cwy.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > Everybody knows what the number of ther EC states is. > > > > That is *not* what I did ask you. You state that it is simply a matter > > of definition how one interprets "to grow" and "number", > > sure. > > > and I asked you > > to provide definitions. Moreover, the number of the EC states is not > > fixed, so you can only state what the number of the EC states is at a > > particular point in time. > > The number of EC states is "the number of EC states". It is simply a > notion which can be equal to a natural number. Again you have provided neither a definition of "number", nor of "grow". Are you unable to do so? In common parlance, but that is not mathematics. In mathematics functions can grow in relation to their argument, but not the entities they denote. > The set of prime numbers does not contain the number 1. But once upon a > time it did contain it. And 2000 years ago it did not contain it > (because at that time 1 was not considered to be a number). That means that ultimately there is not a single "set of prime numbers". There are multiple sets of prime numbers, and it depends on the definition what particular set is intended. So when I am writing about the "set of prime numbers" in connection to D. H. Lehmers work you should be careful to note that I may use the definition he did use. However, when we talk currently about the set of prime numbers in general the most commonly used definition should be understood, the one that excludes 1. More interesting, there are two different sets of positive real numbers in common use. Which one you should assume depends on the forum where you are reading and/or writing. The same for the concept "field". And, indeed, I have used both definition for both at University, depending on the courses. Apparently you do not allow that a term can denote different things in different realms. Mathematical reality is that it *can*. > > > > Wrong. Read my explanation above. I am talking mathematics. > > > > > > Do you really think so? > > > > Yes. > > > That is a matter of definition. You may say: Where I am, there is the > top. Please provide a definition. According to the definitions of set theory a set can not grow. But as you continue to refuse to provide definitions we are left without things to discuss. On the other hand, you always state that you are working within set theory, in that case you should use the definitions of set theory, and there sets can not grow. > > > > > Hrbacek and Jech teach standard set theory including the fact > > > > > that in ZF everything is a set. > > > > > > > > Yes? > > > > > > I remember that you opposed. Now you agree because you have learnt? > > > > No. I still do not agree. The universe in ZF is *not* a set. > > Then there is no universe in ZF. Can you provide a quote from Hrbacek and Jech where they state that everything is a set? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |