Cantor Confusion
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From: Dik T. Winter on 14 Nov 2006 21:05 In article <1163506063.849027.169180(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1163433510.518520.69770(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > > Well-ordering is so easy connected with AC that we can > > > > > state: Well-ordering has been assumed. > > > > > > > > Not in set theory. AC is an axiom that may or may not hold, depending > > > > on the branch you are following. > > > > > > Zermelo considered it as "has to be taken". > > > > Perhaps he did. In modern mathematics there is no such imperative. > > Euclid considered that the parallel postulate "has to be taken". > > This does not mean that in modern mathematics it has to be taken. > > It surprises me that in one case you consider the overthrowing of > > an axiom that was basic for centuries as valid, > > I don't. The parallel axiom is necessary, correct, true in any > Euclidean plane. That is a tautology. But you do not allow elliptic or hyperbolic geometry? If not, why not? > > > Oh, he would rotate in his grave if he heard you. Of course he only > > > assumed those first principles which were true in nature or reality. > > > > What is true about 3 in nature or reality? What is true about a set > > of numbers in nature or reality? > > Cantor thought so, as I just quoted in my last letter to you. And I > think so too. Therefore I ike him by far more than all the modern > mathematicians. I do not ask what you think. Reread my question. What is true about a set of numbers in nature or reality? > > > > Oh, well, do set theory without AC. No problem. There is a number of > > > > people doing it. > > > > > > Zermelo did not belong to this group. > > > > Yes, and so what? Euclid did not belong to the group that does either > > elliptic or hyperbolic geometry. > > But he was a bit suspicious, because he avoided the application of his > 5th postulate as long as possible. Yes, and set theorists are suspicious because they avoid the application of AC as long as possible. And so, in a sense, does Cantor. > I think that the rise of non-euclidean geometries was a big > disadvantage of mathematics, because from then on every guy felt > entitled to create his own system of axioms. Before that time already, I think. And for those alternatives, blame Saccheri, Lobachevsky and Riemann (resp. 1733, 1829 and 1854). But consider the invention of the quaternions by Hamilton in 1843, where he considered a kind of numbers that failed a basic field axiom. And still more basic: complex numbers. Failing the common ordering axioms of numbers. Perhaps first used by Heron of Alexandria (1st century CE). But first actually used in 1545 in the book by Cardano (but that was credited to Tartaglia). Of course, at that time people did not thoroughly think in strict axiomatic systems, but in what they thought were fundamental truths. But in essence there is no difference. -- 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 14 Nov 2006 21:08 In article <1163506214.782141.176790(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > Thank you for your attention. Correction: > > > An initial segment of natural numbers includes its cardinal number. > > > > Further correction. A finite initial segment of natural numbers includes > > its cardinal number. > > Rejected. Why? Yes, I know why. You think that the complete set of natural numbers does contain an un-natural number. I have asked you for a proof based on the basic principles. But you simply refuse to do so. You think it is the case, so it must be the case. So, please, a proof that N contains something that is not a natural number based on the standard axioms of set theory. Once you have done so you will have shown an inconsistency because it is easy to proof that there is no such number. -- 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 14 Nov 2006 21:25 In article <1163507203.468528.168460(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1163453584.773656.46060(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > See <J7E0uw.32v(a)cwi.nl>. "... sets of the first cardinality can be > > counted only through (with the aid of) numbers of the second > > class..." (Abhandlungen, p 213.) > > In German: "w?hrend die Mengen erster M?chtigkeit nur durch (mit > Hilfe von) Zahlen der zweiten Zahlenklasse abgez?hlt werden k?nnen" > > > And that was the quote you denied he explicitly stated. He does not > > state (and I did not write that he did state) that a set of > > cardinality aleph_0 has an omega-th element, but that can easily be > > deduced from the above quote. > > Cantor considered as "Anzahl" which can be determind by "abzaehlen" > simply the ordinal number of a set. But in that quote there is no mention of "Anzahl". There is only mention of "abz?hlen", and if I understand German well, that means, in general, just like in Dutch, the "process of counting". There is one set of the first cardinality (N) that can be counted (i.e. the process of counting) without any reference to w. That is potential infinity, and you did agree. And that is still the case in modern set theory. > p. 213: Durch Umformung einer wohlgeordneten Menge wird, wie ich dies > in Nr. 5 wegen seiner Wichtigkeit wiederholt hervorgehoben habe, nicht > ihre M?chtigkeit ge?ndert, wohl aber kann dadurch ihre Anzahl eine > andere werden. Yes, I know that already. I referred to the place where he states that a set of first cardinality can only be "abgez?hlt" with numbers of the second class. And in my opinion "abgez?hlt" refers to a process, not to a result (the total number). > This does *not* mean that omega is an element of the set. The quote means that during the process of "abz?hlen" we need a number of the second class, and so there is an omega-th element (not an element omega). > It cannot be deduced > that Cantor thought that there was an omegath element. I just did. > > > A set is its contents - and nothing more! There are no ghosts in > > > mathematics. > > > > Eh? > > {1,2,3} is the collection of, and a convenient expression to write that > we are talking about, the numbers 1 ,2, and 3. No, the set *containing* the numbers 1, 2 and 3. > > Sorry, I do not understand what you write here. The set > > {{1, 2}, {3, 4}} > > has two elements. It's cardinality is 2. And each of the elements is > > a set containing two elements. > > Yes, here you are talking about two unordered pairs of numbers. You use > a convenient way to denote that. Nothing else stands behind the {, } > symbols. In particular N = 1,2,3,... = {1,2,3,...}. That does not > prevent to build a set {{1,2,3,...}, 1} with two elements. And omega = {0, 1, 2, ...} (I use { and } here to denote ordered sets.) So you can not replace omega by 0, 1, 2, ... -- 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 14 Nov 2006 21:33 In article <1163510733.868272.250410(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > The first two levels of the tree contain the following initial > > > segments: > > > 0.00... > > > 0.01... > > > 0.10... > > > 0.11... > > > > Note that at every level n of your tree the numbers represented are > > all of the form a/2^n, with 0 <= a < 2^n. So neither 1/3 not 1/5 > > are in any of the finite subtrees. > > They have no finite binary representation. But if they can appear in a > list, then they can appear in a tree. Yes, of course they can appear in a tree. But they do not appear in *this* tree. 1/3 can appear in a ternary tree. 1/5 can appear in a quintary tree. Both can appear in a base 15 tree. And consider the following list (assuming binary): 0.0101010101... 0.001001001001... 0.000100010001... ... a beautiful list, I would think, containing at the n-th position 1/(2^n + 1). > > But that one is not in the tree. If (as I state above) we consider that > > each node also carries the decimals above it (which is equivalent to > > your statement), each rational number in [0, 1) with a power of 2 in > > the denominator is in the completed tree, but no other numbers. > > Even of [0, 1] because 0.111... = 1. > If you have this opinion, I will happily agree, but then you must also > apply it to every binary representation of the reals. The tree is > nothing other than such a representation, a special one. And that is wrong. The tree contains only the set of finite binary representations. > You just stated that there are no infinite strings. A agree. Where did I state that? -- 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 14 Nov 2006 21:44
In article <1163536221.946682.182290(a)h54g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1163469888.057589.117040(a)k70g2000cwa.googlegroups.com> "William Hughes" <wpihughes(a)hotmail.com> writes: > > > So lets count the set of all natural numbers {1,2,3,...} > > > There are no natural number left. So we stop > > > using natural numbers and use ordinals > > > (and to nobody's surprise a few things change). > > > > This is wrong. There is no ordinal needed to count the elements of the > > set of all natural numbers. You can count until you weigh an ounce (;-)) > > but you will never finish. > > In this way you can also count the reals without ever finishing. > Therefore this cannot be the meaning of Cantor's "countable". Indeed. But in the case of countable sets you are sure that you reach each element after a finite number of steps. Not so with uncountable sets. > > Neither the elements you wish to count will > > be exhausted nor the numbers with which you count. I think this is > > potential infinity. > > Correct. > > > On the other hand, when you ask "how many" elements > > there are in N, you need an infinity (and this is, I think, actual > > infinity). > > Correct. > > > But all this hinges quite closely on the semantics of the > > word "count". If seen as a process, you do not need an infinity; when > > seen as the result of a process, you do need an infinity. > > and, above all, you do not need different infinities. This had been > known already before Cantor. And is still true in current set theory. Set theory does not think too much about the "potential" infinity, because it is not so relevant for the theory. On the other hand, other branches of mathematics *do* consider potential infinities. In analysis you will find nothing but potential infinities. And in projective geometry you will again find actual infinities. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |