An uncountable countable set
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From: Dik T. Winter on 25 Aug 2006 22:30 In article <44ef36ae$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Dik T. Winter wrote: .... > > > 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. > > Such that one can specify which finite number of iterations will get one > within a specific finite range of accuracy, gven a specific method of > approximation? It's a limit concept, really, yes? Computer scientist you are? How wrong you are. The very first definition is that a number is computable if there is a Turing machine so you can give it a specific integer n and it will calculate all digits from the first to the n-th. There is no limit concept involved at all. All algebraic numbers are computable, as are a host of non-algebraic numbers (e, pi). -- 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 25 Aug 2006 22:54 In article <44ef3b88(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Dik T. Winter wrote: .... > > > > There is no such specific natural number. It is when we have them > > > > all, but as there is no largest number, this can not be achieved by > > > > taking them one by one. > > > > > > The set of all naturals numbers consists of only natural numbers. There > > > is NO natural number where the count becomes infinite. So there is no > > > point in the set, even if you COULD get to the "last" one, where any > > > infinite set has been achieved. > > > > And there is no point in the set where you have the complete set. Yes. > > Indeed. So what? > > So, if there is no point in the set which can even remotely be > considered infinitely far from the beginning, what makes it actually > infinite? What the set of all finite numbers makes infinite? The axiom of infinitys states that the set of all finite (natural) numbers does exist. From that is is easy to prove that that set is not finite, hence infinite. > If no element of the set can be an infinite number of steps > from the start, you may not be able to find an end. And indeed, when you go step by step you will not get at the end. > But does that mean > it's "greater than" every finite, or only "greater than or equal"? What is the difference? Assuming you mean "aleph-0" when you write "it", it is easily proven that: aleph-0 is greater than or equal to each natural gives the theorem: aleph-0 is greater than each natural. Because: suppose aleph-0 >= each n in N. Now suppose in addition that it is equal to some particular n. Well, n + 1 is in N, and so aleph-0 should also be larger or equal to n + 1. Hence it can not be equal to n. So it is not equal to any n at all. And so aleph-0 > each n in N. > > Indeed. The set of all natural numbers is just sufficient. > > No, it is far too great. If you have a countably infinite number of bit > positions, then you have an uncountably infinite set of strings. Where > bit positions are indexed by the naturals, the naturals are the power > set of the number of bit positions, Wrong. This is plain nonsense. Suppose there are three bit positions. The set of naturals {1, 2, 3} is sufficent to index them. In what way is {1, 2, 3} the power set of the number of bit positions (3)? > > No one is obligated to accept the theory at all. Whether it is proven > > to be "correct" or not, as I have no idea what "correct" in this context > > means. Is Euclidean geometry "correct"? Is hyperbolic geometry "correct"? > > Is elliptic geometry "correct"? > > Ah, now you bring up a prime example. Euclid set down laws for flat 2D > geometry, and questioning those axioms led to new shapes for space. > Accrdingly, the axioms of set theory might work together to describe a > system, but it is not impossible that entirely other systems might arise > from different starting assumptions. And indeed, I never did state the opposite. But if you want to get at a new system, provide axioms, definitions, and whatever. Tell us what axioms to retain and what axioms to reject. And if there are axioms to be rejected, come up with alternatives. > > But what is the case is that if you accept the axioms, you also have > > to accept what follows from the axioms. > > Yes, I understand that, and much to the consternation of some, I don't. Yes, much consternation, I can understand that. So apparently you are accepting the axioms, but not what follows from the axioms. What kind of logic are you using? > Rusin had the gall to tell me that if I don't accept that there are an > infinite number of finite naturals, then I will join JSH and others on > his "kill list". I don't claim that my conclusions are derived purely > from ZFC or NBG, but that there are more fundamental concerns which > contradict both, and that some other prioritization of principles needs > to happen. Proper subsets are smaller. The addition of a single element > needs to be reflected in the size of the set. Infinite values are larger > than finite values. Things like that. Well, you are stating such. So provide a framework. Either within the accepted axioms, or without it. But if you want to remain within the accepted axioms, you should also accept what follows from these axioms. And if you want to go outside, provide your set of axioms. -- 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 25 Aug 2006 23:14 In article <44ef3da9(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Dik T. Winter wrote: ..... > > > Why do you need an axiom for that? Why is it > > > not derivable logically? > > > > Because without the axiom of infinity the set of naturals need not exist, > > and indeed, you can build a completely logical system with the negation > > of the axiom of infinity and with all other axioms remaining. It is > > similar to the parallel axiom in geometry. > > But without an axiom of infinity, it is demonstrable that, given the > axiom of internal infinity (continuity), x<z -> x<y<z, that any finite > interval includes an infinite number of points. Start with the line, and > identify points. There's infinity. Your axiom uses things that are not defined. What is the *meaning* of "x<z"? > > You both do not have an linearisation of the reals. You both are doing > > other things. > > Describe those "other things". How are they "other"? You are expanding the reals in some undefined matter. > > > The second proof of the uncountability of the reals is not about reals? > > > That's news. What was it about, then, in your opinion? > > > > Sorry, I already explained that in previous articles to which you have > > replied. But if you do not read the articles you reply to there is > > something seriously wrong in the discussion. > > It is about digital representation, which is the same as power set, even > in other number bases, which means more than two states of inclusion per > member. Darn. Try to read. Cantor's proof is not about reals, it is neither about digital representations. It is about none of the things you are mentioning. But nevertheless you maintain that it is news that it is not about the reals, while you read what I wrote? > > > Larger than any finite. The set of naturals is as large as, but no > > > larger than, every natural. > > > > That is not a definition, because it makes no sense. "The set of naturals > > is as large as every natural"? > > It is no larger than all naturals That is something completely different again. > > From that: "The set of naturals is as > > large as 1", "The set of naturals is as large as 2". What is the meaning > > of these statements? > > That is when you substitute "every", meaning "each", for "all". Careful. Yes, you should be careful in what you mean, and not use a word that has multiple meanings so that you can be misunderstood. So I will refrase: > > > Larger than any finite. The set of naturals is as large as, but no > > > larger than, all naturals. Is that what you intended? In that case you just stated a tautology. > > > Then I don't know what proof you are talking about. When people say > > > "Cantor's second", they are generally referring to his second proof of > > > the uncountablility of the reals based on the diagonal argument, as > > > opposed to the first, based on an unreachable intermediate value. > > > > But they are wrong. The proof was *not* about the uncountability of the > > reals. The diagonal proof Cantor provided was not about that. It was > > a proof about the things I outlined just above. > > It was about power set and digital representation, which are identical. > It was about symbolic sets. You finally did read it? If so, you really should improve your German. > > > I thought it was clear that I was using a notion of infinite, like WM, > > > from a quantitative standpoint, rather than set-theoretic. > > > > Without definition. > > Greater than any finite. Simple enough? So the cardinality of the naturuals is infinite? -- 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 25 Aug 2006 23:20 In article <virgil-B3927D.12230325082006(a)news.usenetmonster.com> Virgil <virgil(a)comcast.net> writes: > In article <1156501289.435365.119480(a)75g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: .... > > The problem is not indexing but "indexing without covering". > > That is easy to prove impossible for any finite natural number in unary > > representation. And, alas, there are only finite natural numbers. > > I have no idea what "indexing without covering" means, and until it has > a clear definition, I will continue to state that indexing all of them > in any way indexes all of them. What I have been able to find was that with indexing of digit WM means that every digit position can be indexed by a natural number. With covering WM means that all positions to a certain one can be indexed by a natural number. What WM is asserting is that when a number can be totally indexed, it also can be totally covered. Quantifier dislexia disguised. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 26 Aug 2006 01:18
In article <44ef9b7d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > What is the difference between Dedekind infinite and > infinite? A set is Dedekind infinite if it has an injection to a proper subset, and finite otherwise. A set is (non-Dedekind) finite if it has any bijection with any von Nuemann natural, and is otherwise (non-Dedekind) infinite. In ZFC and NBG, they are equivalent definitions. |