An uncountable countable set
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From: Dik T. Winter on 29 Aug 2006 20:00 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..."? -- 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 29 Aug 2006 20:18 In article <1156843043.617422.7230(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1156689126.335154.133150(a)h48g2000cwc.googlegroups.com> muecke= > nh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > > > The first part of his first proof shows that a complete ordered= > field > > > > > > has cardinality larger than the natural numbers. In his proof = > he did > > > > > > not rely an any properties of the reals other than that they fo= > rm a > > > > > > complete ordered field (he uses reals to exemplify). > > ... > > > > Note what I wrote: "in his proof he did not rely on any properties of > > > > the reals other than that they form a complete ordered field". What > > > > other properties of the reals did he use? > > > > > > He had no field and he used no filed. What is a field without the > > > axioms of the field? Nothing. Cantor disliked axioms if he did not hate > > > them. His only concern were numbers, numbers, numbers and their > > > *reality*. No fields and no axioms. > > > > Perhaps. I thought we were discussing current set theory. And not what > > in some time long ago was et theory. He may not have liked them, but in > > current terminology "the first proof shows that a complete ordered field > > has cardinality larger than the natural numbers". > > In the terminology of the next century it may show even other things, > more general perhaps, with even more insight. I was discussing what > Cantor did. You were accusing me that I had not understood or misquoted > him. Please reread the history of this discussion. I did not accuse you of anything in this instance, only on misreading what I wrote. At some stage I responded to somebody else (I think it was Virgil) that in the first proof Cantor showed that complete ordered fields were not countable. You accused me of misinterpretation. The only case where I accused you of misquoting was when you left out the first sentence of a paragraph that would be revealing. But, see below. > > > Aus dem in =3DA7 2 Bewiesenen folgt n=3DE4mlich ohne weiteres, da=3DDF > > > > Quoting again only in part. I will quote the translation I gave, with > > annotation of the complete paragraph, not only the part you like to quote: > > In the article, titled: "?ber eine Eigenschaft des Ombegriffs aller > > reellen algebraischen Zahlen (Journ. Math. Bd. 77, S. 258) [hier > > S. 115], can for the first time be found a proof of the theorem > > that there are infinite sets that are not in bijection with the set of > > natural numbers 1, 2, 3, ..., v, ..., or, as I say in general, do not > > have the same cardinality as the natural numbers 1, 2, 3, ..., v, ... . > > So apparently he is thinking that his first article proves the theorem > > that there are sets that are not in bijection with the natural numbers. > > Which is true. > > Of course: The set of the reals and the set of the transcendental > numbers are such sets. So he has the right to speak generally about > "sets". Which mathematician would not like to generalize his theorems? But note what he now claims the theorem actually *is*. > > From what has been proven in section 2 follows that e.g. the > > set of real numbers in an arbitrary interval can not be put in > > a sequence w_1, w_2, w_3, ..., w_v, ... . > > So he is now thinking that that proof was just an example for such a set. > > Which it is. The "beispielsweise" is telling. > > It is however possible to construct a much simpler proof for > > that theorem that is independent from the observation of irrational > > numbers. > > Independent of the *necessary use* of irrational numbers. Where in the German do you find "necessary use"? "ein viel einfacherer Beweis liefern, der un?bhangig fon der Betrachtung der Irrationalzahlen ist." > > And the "that theorem" (in German "jenem Satze") can, in my opinion only > > refer to the theorem mentioned in the first sentence in this paragraph. > > That theorem the proof of which depends on irrational numbers. Nope. That theorem is the theorem he stated in extenso in the first sentence. -- 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 R Tribble on 30 Aug 2006 16:14 >> But since no natural requires an infinite bit string, that is irrelevant. Tony Orlow schrieb: >> If no natural requires an infinite bit string, even the very largest, >> and all bit positions are included in it, then no infinite set of bit >> positions is required. > mueckenh wrote: > Tony, please drop the "very largest". The rest is ok. If no natural > requires an infinite set then all naturals together do not require an > infinite set. If the contrary is asserted (and if infinity is a number > larger than any finite number), then we may ask which natural is the > first such that all of its predecessors require an infinite number of > bit positions. There is no natural with an infinite number of bits. Every natural is a finite binary string. There is no natural with an infinite number of predecessors. Every natural has an infinite number of successors, however. (Which agrees with what you just said to Tony about there being no "very largest" natural.) It is truly breathtaking that there are people who cannot grasp the elementary concept that if there is no largest finite natural then there must be an infinite number of them. Even children understand that there is no biggest number (because you can always add one to any number you think is the largest), and therefore "numbers never end" (to quote my six-year old daughter).
From: David R Tribble on 30 Aug 2006 16:29 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? 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?
From: Virgil on 30 Aug 2006 17:16
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. |