An uncountable countable set
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From: Virgil on 2 Sep 2006 12:57 In article <1157194212.124804.231590(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > I did already tell you which edges belong to the path 1/3. The nodes > > > can be enumerated like the edges. > > > > Yes. But you still only have terminating paths. At each level all paths > > are terminated by a node. And so 1/3 is not in the tree because there > > is no terminating edge and node. > > Then such numbers like 1/3 do not exist (in that representation). But > the same holds for Cantor's list. > > > > Oh, well. As in mathematics the reals require a construction process, so > > also your tree requires a construction process. In mathematics the reals > > are constructed from the rationals (and I know at least four methods to > > do that, that can be shown to give equivalent results). And the rationals > > are constructed from the integers. I asked you before, but you never did > > reply. Do you know how the rationals are constructed from the integers in > > mathematics? More basic, do you know how arithmetic on naturals is defined, > > based on the Peano axioms? > > More to the topic: Do you know how Cantor's diagonal is constructed > from a list of reals? And how this list is constructed? Cantor's "diagonal" in its original form applied only to the set of all binary sequences, i.e ., infinite sequences whose terms were from a set of two objects, and not to real numbers of decimal digits. This is essentially the same as the set of all functions from N to {0,1}, where N is understood to be an endless set of natural numbers. His theorem says that the set of all such functions could not be listed sequentially without leaving some out. > > What is the difference between my tree and Cantor's list. Lists are sequences which do not branch. Trees branch. > > > Either you agree that all the edges of my tree are countable or you > > > agree that Cantor's diagonal is uncountable. Or you state at least how > > > many infinite path in my tree you would consider to have completely > > > countable edges. How is one object, Cantor's diagonal sequence so much more than one object as to be uncountably many objects? > > The problem is the following: You assert that the digit positions of > 1/3 are countable as well as the levels of my tree (all of them), but > you deny that the edges at these levels are not all countable. That is > wrong because for every n-th level you count, the number 2^n of edges > can also be counted. There is no limit. It is only the set of individualy infinite paths in that tree that are uncountable, The set of all edges and the set of all nodes are both countable.
From: mueckenh on 2 Sep 2006 16:08 Dik T. Winter schrieb: > In article <1156842732.043353.163480(a)74g2000cwt.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > Cantor states (analogously) that *all stairs exist*, that the width of > > > > all ot them is L, but that none of them has hight L. Width [0, 1] and > > > > height [0, 1). > > > > > > A quote please. But indeed, the width of all of them is L, but there is > > > none of them with width L. And the heighth of all of them is L, but there > > > is none of them with heighth L. So width [0,1] and heighth [0,1]. That > > > is, if you define both as the smallest box containing the completed stair. > > > If you define both as the largest numbers that can be obtained, you get > > > width [0,1) and heighth [0,1). > > You are trying to make it difficult, using UTF-8. But this one takes the > cake. What is UTF-8? > > I agree with you. But Cantor said (Werke, p. 409): "So stellt uns > > beispielsweise eine veränderliche Größe x, die nacheinander die > > verschiedenen endlichen ganzen Zahlwerte 1, 2, 3, ..., v, ... > > anzunehmen hat, ein potentiales Unendliches vor, wogegen die durch ein > > Gesetz begrifflich durchaus bestimmte Menge (=EF=81=AE) aller ganzen > If "=EF=81=AE" is UTF-8, the Unicode character is U+F06E, which is in the > private area of characters. I have no idea what that symbol stands for, > so I will modify it to N. > > endlichen Zahlen N das einfachste Beispiel eines aktual-unendlichen > > Quantums darbietet. > > Again translated (why do you post so much German in an English speaking > newsgroup while you should know that most readers are not able to read > German?): Because I have the German text available and because I do not want to be blamed of mistranslating. > Cantor: So while a changing quantity x that successively takes the > various values of finite numbers 1, 2, 3, ..., v, ... , is a > potential infinite, on the other hand, a through the axioms completely > determined set (N) of all integral finite number is an example of an > actually finite quantity. Not through the axioms, but through "a law" (ein Gesetz). > Nice that you found the quote I have alluded to, and that you did deny > of existing, but that I could not find back. > > What Cantor is stating here (and I did already indicate that in an > earlier response), is, translated to current set theory: > The set N is potentially infinite, No. The changing quantity is here a variable. >the size of N is actually infinite. > (In current terminology a set is not a quantity.) Cantor's changing quantity is a variable. A set is never potentially infinite according to Cantor. Regards, WM
From: mueckenh on 2 Sep 2006 16:09 Dik T. Winter schrieb: > In article <1156842898.805375.169610(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > You gave an example how a number of the form 0,111...1 with n digits > > > > indexes the n-th digit but does not cover all digits with m =<n ??? > > > > > > No, because that does not exist. > > > > How then can you believe and assert that the number 0.111... which has > > *only finitely indexable positions* could be completely indexed but not > > covered. > > Because there are infinitely many finitely indexable positions. Do you > see the way in which the first number you give "0.111...1" differs from > the second number you give "0.111..."? For indexing we have exactly the same question. Regards, WM
From: mueckenh on 2 Sep 2006 16:12 David R Tribble schrieb: > Dik T. Winter schrieb: > >> But indeed. If the ordinality of a set is infinite (w or larger), then > >> so is its cardinality (aleph-0 or larger). On the other hand, if the > >> cardinality of a set is infinite (aleph-0 or larger), and if it can be > >> well-ordered, so is its ordinality after well-ordering (w or larger). > > > > mueckenh wrote: > > As long as the set of natural numbers includes only finite numbers, its > > cardinal number is also finite. > > What is that cardinal number? Do you have a name for it? The set is potentially infinite. It has no cardinal number in the sense of set theory. All we can attach to it is the number of elements known or existing. Disregarding physical constraints we can assume that the elements of the set of natural numbers are counted by the largest natural number temporarily known. With respect to physical constraints the number of elements is less than 10^100. > > Tony calls it "alpha", the cardinality of the finite but "unbounded" > set of all finite naturals, and also the largest natural. It appears > to have the curious property of being a finite natural but having > no finite successor. He's never provided an estimate of how > large it is, though. Perhaps you have a better idea of this? Sorry, there is no largest natural and no natural is infinite. Regards, WM
From: mueckenh on 2 Sep 2006 16:13
Virgil schrieb: > In article <1156363517.395289.207370(a)i42g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Tony Orlow schrieb: > > > > > > > > Ordinals and cardinals are necessities if we want to talk about > > > > set "order" and "size" in any kind of logical, well-defined way. > > > > That is wrong. We can talk about finite sets and about infinite sets > > which all have the same magnitude by the measure of intercession > > instead of bijection. > > "Intercession"? What sort of measure is that? > > > > > > > > > What do you call the "size" of a countable set with no end? > > > > You don't have a name for it, do you? > > > > > > No, I find focused concentration on the Twilight Zone, the "boundary" > > > between finite and infinite, to be a rather fruitless exercise. There is > > > no such distinct boundary. > > > > > > > > Because there is no boundary at all. All sets are finite, but some are > > without end. > > Since "infinite" commonly means endless, that is a bit tricky. > > > > > Those are called potentially infinite. All of them have > > the same magnitude if measured by intercession. > > > > Definition: "A and B intercede": An order can be defined such that: if > > b_1 and b_2 are elements of set B, then a at least one element a of set > > A lies between them in this order, and if a_1 and a_2 are elements of > > set A, then at least one element b of set B lies between them in this > > order. > > > > > Example: Rational and irrational numbers intercede in the natural order > > by magnitude. > > But when one well orders the rationals, one apparently changes the size > of the set without adding or removing a single member, since it can now > be made to intercede with the naturals. > > I find any measure of set size that depends on order to the extent that > reordering without adding or deleting any members changes the size, to > be ridiculous. All infinite sets (of finite numbers) intercede each other. There is only one equivalence class. It is not depending on order. Regards, WM |