Cantor Confusion
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From: mueckenh on 10 Dec 2006 14:16 Eckard Blumschein schrieb: > > Then why do you disagree with Cantor's results? > > This is a good question. I disagree with those of his results which are in error. Cantor knew how to distinguish potential from actual infinity, that is more than most discussers here can achieve, but he was not infallible. Regards, WM
From: Virgil on 10 Dec 2006 14:19 In article <1165758455.659914.107080(a)j72g2000cwa.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Virgil schreef: > > > In article <1165695789.274304.75780(a)80g2000cwy.googlegroups.com>, > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > > > stephen(a)nomail.com schreef: > > > > > > > functions, etc. as all of those things can be modelled with set theory. > > > > > > The topic of functions has been handled separately on my web page: > > > > > > http://hdebruijn.soo.dto.tudelft.nl/www/grondig/natural.htm#fd > > > > > > In a nutshell: the mainstream definition is narrow-minded because the > > > whole notion of _TIME_ is lacking. > > > > Position, velocity and acceleration are specifically expressed as > > functions of time, so I have no idea of what HdB is talking about. > > Huh, no. Let f(x) = 2.x then time is involved with multiplying x by 2. Does that exempt velocities and accelerations from being time dependent? > > > > Absolute rigour is a phantom, even in mathematics. > > > > Absolute lack of rigor is chaos, especially in mathematics. > > Absolute lack of rigor is chaos. But lack of absolute rigor is nice. Is that an attempt to make nice?
From: mueckenh on 10 Dec 2006 14:21 Dik T. Winter schrieb: > > > 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. > > > > In the same way as the first few digits of a real number represents a > > number. 3.1, 3.14, and so on represent numbers in some way. But that is > > not at all important or interesting for the tree argument. > > So why did you state that I erronously believed that nodes represent > numbers? Because you erroneously did. (The real numbers in the tree are all represented by infinite paths like 3.1000... 3.14000... etc.) > > > > > Not at all! I represent numbers by standard binary notations. > > > > > > It is using the limits where you are doing something non-standard. > > > > I do nothing. The tree cares that even in the limit the number of paths > > cannot become uncountable. 2^n remains the cardinal number of a > > countale set, even in the limit n --> oo. That's why I devised the > > tree! > > And it is exactly that what is wrong. For each finite n 2^n is the > cardinal number of a countable set (even of a finite set), that does > not make something like that also true in the limit. What limit are you talking about? My statement is true for each finite level n. There are no others. > It is easy > enough to construct a bijection between the natural numbers and the > edges, because the edges are countable. After all you have grasped that too? Until now you denied. > Contrary to what you write > elsewhere, you have *not* constructed a surjection from the edges to > the paths. If you think you did that present us with an edge that > maps to 1/3, and show how the mapping is constructed. Please give me all the bits of 1/3. Then I will show the bijection. ============================= > It is easy enough to construct a surjection from the edges to the > binary rational numbers with terminating expansion. But indeed, WM > does not show a surjection at all. Indeed you are unable to understand fractions? Regards, WM
From: mueckenh on 10 Dec 2006 14:25 Dik T. Winter schrieb: > In article <1165493929.872320.230630(a)n67g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > Apparently you do not allow that a term can denote different things in > > > different realms. Mathematical reality is that it *can*. > > > > That's all depening on the definition. I advocate the idea that sets > > can change and numbers can grow because the common (contrary) > > definition leads the direct way to the (false) result of actuallity we > > observe in modern set theory. There is no room for potential infinty. > > But you do not provide a (mathematical) definition about what it means > (you almost never give mathematical definitions), so who can those > things be talked about mathematically while making sense? You do not know what a defintion is, unless it is given in the very primitive language of set theory. It is not suitable for potential infinity. > > > But I am not surprised to hear this harsh critic. With their words and > > with their notions modern set theory will perish. > > You think so. I think modern set theory will survive you. I am sure there will always people who believe or pretend to believe this stuff. So you are right. But the great majority will soon recognise that it is nonsense. > > > So the only objects with which we are concerned from now on > > are sets. > > So they do not state that everything is a set. Thing and object is used synonymous. But I would have been very surprised seeing you to admit an error. Ask people who know set theory better than you (in case you believe that such people exist). They will tell you: Everything in ZFC is a set. Or look in one of many aknowledged books. You will find teh sentence, in ZF everything is a set. I will not bother to take a book from the bibliothy in order to prove you wrong. We both know that you are wrong, and those who know ZF know that too. > > > But this point of view is also entertained in many other modern books: > > In ZFC everything is a set. > > Again, no. Or do you not see the difference between "all objects" > and "everything"? A theorem obviously is not an object, but it is > contained in "everything". And also the universe is not an object > in ZF set theory. Everything in ZFC is a set. The universe is not in ZFC. It would be the set of all sets. Regards, WM ======================================= > As you never show your statements with quantifiers in a proper mathematical > fashion you do apparently not see that the quantifiers are about infinite > sets. Please try to learn: EVERY LINE IS A FINITE SET. And forget your handwaving about "infinite sets". O course you need this escape ibn order to avoid admitting that there can be no infinite set of finite elements. But I do not agree to accept this esacape. It is sufficient to consider only lines (every element of the diagonal is in a line). And every line is finite. Therefore, if every element of the diagonal does exist, then every element of the diagonal is in one line. This leads to the result that not every element of the diagonal does exist. Regards, WM
From: mueckenh on 10 Dec 2006 14:27
stephen(a)nomail.com schrieb: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > Bob Kolker wrote: > > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > >>> You cannot imagine the integer [pi*10^10^100]. > >> > >> That is not an integer, dummkopf. It is an irrational real number. > > > Dummkopf? Who? Doesn't "[ .. ]" stand for the "floor" function? > > > Han de Bruijn > > Not anywhere with which I am familiar. As [pi*10^10^100] was explicitly denoted as an integer, even a mind unfamiliar with the conventional meaning of "Gauss-brackets" should have been able to recognize their meaning. Regards, WM |