An uncountable countable set
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From: Virgil on 29 Aug 2006 13:47 In article <1156842504.966630.131290(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > 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. It is not how big the numbers are but how many of then that is at issue, and there are still more than any finite number of them, which in any reasonable sense means infinitely many of them. > > > 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. While there is no "acual infinity" as a *member* of the set, that is not what is being discussed. > > > > > Your example shows very clear that either both LUBs have to be included > > > - or none. In my opinion we choose none. But whatever you choose, you > > > cannot avoid to see the symmetry, can you? > > > > I see the symmetry, I also choose none. And I think I have stated that > > already. > > Fine. That means there is no actual infinity. No, that means that there is no actual "Mueckenh". > > > But if there are infinitely many naturals then the infinite sum is > > > necessary. > > > > Not at all. Why do we need it? > > 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), "Infinity" is not a thing, as there are too many different sorts of infinteness to be any one thing. There are types of "numbers" which do not have the property of being finite. > then there is at least one infinite sum > of 1's (one from each natural). Why? Such extraordinary claims require extraordinary proofs, and "Mueckenh" cannot give any proof at all. > > > > 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, Who says that there are even numbers greater than alef_0? Name one. > > 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). > > As long as the set of natural numbers includes only finite numbers, its > cardinal number is also finite. Except when it is not. And the cardinality of the set of naturals is larger ( in the Cantor sense of cardinality) than any natural. I.e., there is no natural large enough to be the cardinality of the set of all naturals. > > Regards, WM
From: Virgil on 29 Aug 2006 13:56 In article <1156842646.101182.155640(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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)? Or, what is much more likely than the inconsistency of something so thoroughly vetted as standard set theories, is the likelihood that "Mueckenh"'s theory is inconsistent. > Either you agree that all the edges of my tree are countable or you > agree that Cantor's diagonal is uncountable. The set of edge is countable, but, for an infinite binary tree, the set of paths is not.
From: Virgil on 29 Aug 2006 13:59 In article <1156842898.805375.169610(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Indexing does not require "covering" in any sense except that each object to be indexed is "covered" by having an index assigned to it.
From: Virgil on 29 Aug 2006 14:06 In article <1156843043.617422.7230(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1156689126.335154.133150(a)h48g2000cwc.googlegroups.com> > > mueckenh(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 form > > > > > > 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. Cantor required of the his set of reals, with its usual arithmetic and order properties, all those properties that we now collect under the rubric of a complete Archimedean ordered field, even though he did not call it that. Or does "Mueckenh" claim that there is some property of a complete Archimedean ordered field that Cantor did NOT assume his notion of the reals had?
From: Dik T. Winter on 29 Aug 2006 18:29
In article <1156689269.210304.49390(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > So you can compute all solutions of a polynomial equation even of > > > higher than fourth degree in finite time? I doubt that. > > > > I did not state that. I said that the numbers were computable, where > > I use the mathematical sense of computable. > > And that means you can compute a number even if you need infinite time. Rather, you can compute the number to asome (arbitrany finite precision in finite time. > Therefore in the mathematical sense each algebraic number is > computable. Right. > (In reality it is not.) That is why I said exlpicitly *in the mathematical sense*, and when writing in this group the mathematical sense is generally understood. > But even in mathematical sense not > *all* algebraic numbers are computable. *Each* one is computable. In mathematics that means that *all* are computable: for all algebraic numbers there is a turing machine that shows that number to be computable. But this does *not* mean: There is a turing machine that shows that all algebraic numbers are computable. > Should you be able to quote a serious scientist who asserted that the > list of all algebraic numbers was computable? (You would prove this > scientist unserious.) Cantor? But what it means for a list to be computable is that given an (arbitrary) natural number n, you can determine the number it points to in finite time. That is all that it entails for a list to be computable. And I would think most mathematicians consider that a list of algebraic numbers is computable. So you apparently think that mathematicians are not serious scientists. So be it. On the other hand, you are not a serious mathematician. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |