Cantor Confusion
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From: David Marcus on 7 Dec 2006 19:34 mueckenh(a)rz.fh-augsburg.de wrote: > 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! The number of paths in a tree of height n is 2^n. Are you saying that the number of paths in the infinite tree is lim_{n -> oo} 2^n? -- David Marcus
From: David Marcus on 7 Dec 2006 19:38 stephen(a)nomail.com wrote: > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > stephen(a)nomail.com wrote: > >> Nobody but you has talked about "growing" sets. Sets, like numbers, do not > >> grow. You, like many other people who do not understand set theory, > >> think of sets as mutable objects, that change as we perform operations > >> on them. This is akin to thinking that numbers change when we perform > >> addition. If I add 3 to 7, neither 3 or 7 changes. > > > It is such an odd belief. Why use a set for something that a function is > > naturally for? I don't really understand why cranks insist on using sets > > for everything, while at the same time insisting that sets are useless > > or illogical or whatever. > > I do not think it is that odd. In everyday usage, the word "set" is used > to denote something that changes. But then again, so is the word "number". > The number of people in a room may change, but that does not imply that > a specific number, such as 5, changes. Most people seem to have an abstract > enough concept of number that the common usage does not confuse them. However > they do not apply this abstraction to sets, so if someone says the set of people > in the room changes, they think a specific set changes. That makes sense. And, most people have no understanding of the function concept, so they don't see its utility or pervasiveness. -- David Marcus
From: Dik T. Winter on 7 Dec 2006 19:35 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? > 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 thikn modern set theory will survive you. > > Can you provide a quote from Hrbacek and Jech where they state that > > everything is a set? > > Pae 2: In this book, we want to develop the theory of sets as a > foundation for other mathematical disciplines. Therefore, we are not > concerned with sets of people or molecules, but only with sets of > mathematical objects, such as numbers, points of space, functions, or > sets. Actually, these first three concepts can be defined in set theory > as sets with particular properties, and we do that in the following > chapters. So the only objects with which we are concerned from now on > are sets. So they do not state that everything is a set. > 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. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 7 Dec 2006 19:40 stephen(a)nomail.com wrote: > David Marcus <DavidMarcus(a)alumdotmit.edu> 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. > > > Perhaps he meant the brackets to indicate the greatest integer less than > > or equal to the bracketed quantity. Although, I don't find any > > difficulty in imagining that integer. > > You only imagine that you can imagine that integer. :) I imagine that you are correct! -- David Marcus
From: Dik T. Winter on 7 Dec 2006 19:49
In article <1165495265.790747.266270(a)f1g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1165492756.322548.255040(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > In article <1165421463.339178.48680(a)j44g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > > The set in the quantifiers you are using is not finite either. The > > > > quantifier are not over a single line, but over the set of natural > > > > numbers. > > > > > > For finite natural numbers, we have finite lines only. It is not the > > > question of a single line. Every line is finite. Therefore there is no > > > line where quantifier reversal could not be applied. > > > > But the quantifier reversal is *not* applied to individual lines, it > > is applied to the set of natural numbers. > > No. It is, refer back to the first time that I properly displayed it with quantifiers: forall{n in N} thereis{m in N} {index m is not in line n} which is true, vs: thereis{m in N} forall{n in N} {index m is not in line n} which is false. The quantifiers are clearly over N, which is an infinite set. It was in response to your: > 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. 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. Note that what the quantifiers really are about (the part between the last pair of braces) is actually irrelevant. And you apparently disallow that inversion when that statement reads {m > n}. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |