An uncountable countable set
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From: Dik T. Winter on 31 Aug 2006 07:59 In article <J4u33q.LCA(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > In article <virgil-026B14.15185630082006(a)news.usenetmonster.com> Virgil <virgil(a)comcast.net> writes: > > In article <1156363640.845840.187460(a)75g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > The cardinal number aleph_0 is infinite while the ordinal number > > > remains finite in order to have infinitely many finite numbers. > > > > Nonsense. > > > > > > > For my part, I agree that the set of finite > > > > > naturals is finite, though unbounded, > > > > > > > > In that case you are not using standard mathematical terminology. I > > > > have no idea what a finite but unbounded set is. > > > > > > That's why you cannot understand mathematics. You fall back behind > > > Cantor. He knew it. > > > > Actually, Cantor knew better than that. > > Actually Cantor made an error. It was in one of his first papers where > he wrote that you also could sort of count with transfinite numbers, and > you the numbers you could count where those of the same class. Found it. First some definitions used: the first cardinality is aleph-0, numbers of the first class are the finite numbers, numbers of the second class are the ordinal numbers belonging to aleph-0. Now the quote (in translation) on page 169 of the "Gesammelte Abhandlungen": ... each set of the first cardinality is countable with numbers of the second class and only through them ... The crucial word here is "countable". What does it mean? If you give it a strict interpretation the sentence is false: the naturals (of the first cardinality) are (in their standard order) "countable" by the numbers of the first class... I think this is the source of the confusion. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 2 Sep 2006 06:42 Dik T. Winter schrieb: > > Quantifier dyslexia is the assumption that there is a number that > > counts all natural numbers, i.e. which is larger than all natural > > numbers. > > That is not quantifier dyslexia. That is an error by Cantor I already > did comment on. That is, when he writes that to count all natural > numbers you need numbers of the second cardinality (or was it the first? Die erste Zahlenklasse (I) ist die Menge der endlichen ganzen Zahlen 1, 2, 3, ..., auf sie folgt die zweite Zahlenklasse (II), bestehend aus gewissen in bestimmter Sukzession einander folgenden unendlichen ganzen Zahlen; erst nachdem die zweite Zahlenklasse definiert ist, kommt man zur dritten, dann zur vierten usw. > I disremember what he called aleph-0), he was wrong. To count all natural > numbers you need only natural numbers. Great! So we have complete consensus now. Excuse me that I did overlook your previous statement concerning this. > On the other hand, the "size" > of the set of natural numbers is not a natural number. It is not a number at all. > > Note, you may critique Cantor's set theory, but that was set theory in > its infancy. It still did contain inconsistencies and errors. Since > that time quite a bit has been developed and corrected. In particular by intermingling the different meanings of infinity. > > Right. Because 0.111... has only finite digit positions we have: > (1) *each* digit position can be indexed > (2) *each* digit position can be covered > but not > (3) the number can be covered. Your "each" means in symbols of logic: "A = (for) all". The number is nothing than all of its digit positions. Therefore your statement is a self contradiction. > Because (3) would mean that there is a number that covers *all* digits, > and as that number also indexes a digit position, there is a last finite > digit position. But that is not the case. Correct. Therefore 0.111... would imply a self-contradiction if it existed. Conclusion: 0,111... does not exist. Regards, WM
From: mueckenh on 2 Sep 2006 06:45 Dik T. Winter schrieb: > In article <1156842504.966630.131290(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > [ talking about {1, 2, 3, ..., w} ] > > > > Let us remove omega from that set. What is the resulting cardinality? > > > > Not an actually infinite one, because without omega there are only > > finite natural numbers. > > But what then? It is a potentially infinite set, as described by Peano and by all the mathematicians before Cantor. > > > > As there is a bijection between that set and the set including omega, their > > > cardinalities are the same. > > > > No. Without omega there is no actual infinity. > > But there is a bijection. And two sets that are in bijection with each > other have the same cardinality. You may think that is a stupid definition, > but nevertheless, it is a definition that works. Look at my last post concerning the final proof that 0.111... with actually infinitely many digits does not exist. > > You should know that the n-th natural number is defined by the sum of > > the number n-1 plus 1 (Peano). This can be retraced to the sum of n > > 1's. What is there to be defined? If there are infinitely many naturals > > (and if infinity is a number), then there is at least one infinite sum > > of 1's (one from each natural). > > Assuming that that sum terminates. Which it does not as there is no last > natural number. Not even set theory makes that a terminating sum. And > this is regardless of whether infinity is a number or not. Cantor defined aleph_0 in this manner: "Da aus jedem einzelnen Elemente m, wenn man von seiner Beschaffenheit absieht, eine "Eins" wird, so ist die Kardinalzahl [von M] selbst eine bestimmte aus lauter Einsen zusammengesetzte Menge, die als intellektuelles Abbild oder Projektion der gegebenen Menge M in unserm Geiste Existenz hat." But if you think he was wrong, there is no need to discuss this topic. > > > > OK, I kind of understand. I do not state that infinity is a number > > > aleph_0. There exists a host of infinities. And aleph_0 is the > > > infinity that gives the equivalence class of sets that are in > > > bijection with the set of natural numbers; the canonical set with > > > cardinality infinity. > > > > If aleph_0 > n for n e N, and if there are even numbers x > aleph_0, > > then aleph_0 is a number. Cantor stated that. If you do not do so, then > > we are in agreement in that point. > > You may call it a number, or not. I prefer to call it a cardinal number. > I prefer not to use the word "number" singly, unless there is context > that shows what kind of numbers I am talking about. So I may be talking > about the numbers in the ring Q(sqrt(-3)), or even about the integers in > that ring. Or about the 5-adic numbers, or the Cayley numbers, but > always with context. I think that in his early papers he indeed called > them "Zahlen", but in later papers he did call them "Mächtigkeit". If > you read his papers you may have found that he changes terminology > between papers. Understandable, because it was an early development. > On the other hand, you will see letters by Kronecker where he calls > the elements of Q(sqrt(-3)) or somesuch "Zahlen" (while he already > much earlier tried to remove all non-integers from proofs). My point is only that: aleph_0 is not in trichotomy with natural numbers and other cardinals. > What is the cardinality of the set of all finite natural numbers? It has none. > But > see above. You assert that it is finite. No. Regards, WM
From: mueckenh on 2 Sep 2006 06:50 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? > > > > and there is an > > > edge that terminates at a node terminating the path. I meant the paths > > > during the intermediate process of construction. > > > > There is no process of construction. If you have difficulties to > > comprehend that: it is the same as with Cantor's list. The tree is > > defined once and for all. That's it. > > Wrong. But I am not going to explain that again. What is the difference between my tree and Cantor's list. There was no explanation and there is none, because there is no difference. > > > All real numbers of the interval [0, 1] are there, some of them even > > twice. > > Many of them are not. Your tree only contain numbers with a finite binary > expansion. 1/3, 1/5 and 1/7 are not among them. But Cantor's list only contains numbers with an infinite binary expansion? In particular his diagonal construction. > > > > > How many paths are required to obtain an > > > > uncountable set of edges, according to your opinion? > > > > > You seem to misunderstand the tree. If the diagonal is in Cantor's > > list, then 1/3 is in my tree. Can you give a reason why it should not > > (other than that then set theory is inconsistent)? > > I have explained already many times, but you are not willing to listen. Please give a reference, or better: copy and paste your explanation. > > > You recently mentioned an interesting aspect: The algebraic numbers, > > i.e., the polynoms are countable, because they are finite. And, > > therefore, you wanted only to count the finite segments of paths in my > > tree. I oppose, unless you agree that the 1's in 0.111... also are > > uncountable. You see, the problem is the same: We can count finite > > segment but not infinity. > > You are using a pretty strange terminology. You can count the algebraic > numbers just because the polynomials remain finite (but there are > infinitely many of them). You can count the finite paths because they > are finite (but there are infinitely many of them). You can count > the digits because each digit position is finite (but there are infinitely > many of them). What inconsistency? You can count the levels of my tree just because they remain finite (but there are infinitely many of them). The number 1/3 has only finite digit positions at finite levels of my tree. Where should any uncountable edge appear? > > > 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. > > This question makes not sense to me. 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. Regards, WM
From: Virgil on 2 Sep 2006 12:42
In article <1157193952.357765.296410(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1156842504.966630.131290(a)h48g2000cwc.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > [ talking about {1, 2, 3, ..., w} ] > > > > > > Let us remove omega from that set. What is the resulting cardinality? > > > > > > Not an actually infinite one, because without omega there are only > > > finite natural numbers. > > > > But what then? > > It is a potentially infinite set, as described by Peano and by all the > mathematicians before Cantor. Does removing 1 from {1, 2, 3, ..., w} change it from an actually infinite to a potentially infinite set? If not then rearrange it as { w,2,3, ..., 1} and we can now remove w and the result is still actually infinite. "Mueckenh" 's silly notion that adding or removing one object from a set can flip it between potentially infinite and actually infinite is both potentially and actually idiotic. > > > > But there is a bijection. And two sets that are in bijection with each > > other have the same cardinality. You may think that is a stupid > > definition, > > but nevertheless, it is a definition that works. > > Look at my last post concerning the final proof that 0.111... with > actually infinitely many digits does not exist. It does in the world of mathematics, which is the only one that counts. Pun intended. > My point is only that: aleph 0 is not in trichotomy with natural > numbers and other cardinals. Which natural is it not greater than, and which other transfinite is it not less than according to the Cantor definition of "<=" for cardinalities ??? > > What is the cardinality of the set of all finite natural numbers? > > It has none. The "Mueckenh" does not choose to recognize one is not proof. The bijections of the set of all finite natural numbers with other sets define its cardinality. |