An uncountable countable set
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From: MoeBlee on 25 Aug 2006 13:52 Tony Orlow wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > >> Albrecht wrote: > >>> We must conclude that we can only say: we can find infinite many points > >>> on a line. This is a potential infinity. > >> Do you have axioms for your mathematical theory of potential but not > >> actual infinity? > > > > Five Peano axioms. > > > > Regards, WM > > > :) Too bad a smiley is not enough for a definition from Peano axioms of 'potential infinity'. MoeBlee
From: Tony Orlow on 25 Aug 2006 14:03 Dik T. Winter wrote: > In article <44eef4aa(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > Dik T. Winter wrote: > ... > > > > And, with the addition of which natural number to the set does the set > > > > achieve this unnatural size? > > > > > > 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? 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. But does that mean it's "greater than" every finite, or only "greater than or equal"? > > > > > > Then show that the set of all natural numbers does not have > > > > > cardinal number aleph-0 and ordinal number w. A proof please. > > > > > > > > I have already shown how the set of bit positions in the binary > > > > naturals has no cardinal or ordinal that can be assigned to it. > > > > > > Yes, but your showing was not a proof. > > > > Yes, it is. A finite number of positions is too few, so it must be > > infinite, but the smallest infinite is too large. > > 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, each with a unique set of 1's. For which set is the set of naturals the power set? None in the standard theory. > > > So, you have a > > contradiction when trying to assign a cardinality to this sequence of > > bits. There is no cardinality for K which does not produce a contradiction. > > But there is. It is the cardinality of the naturals, and that is something > that *can* be assigned. > > > > Sorry, that is not current understanding, that is using the definitions > > > and terminology. Understanding may change, but within the definitions > > > and the terminology any unbounded set will remain infinite. > > > > FIne, the current "system" then. > > Right. You might come up with a different system where that is not the > case. It's almost ripe. Gestation periods of new species are hard to determine. I'm working on foundations, as I work on other things as well. :) > > > > > Using that system, the phrase is senseless, but that system is not the > > > > real universe. It's a concoction. > > > > > > Well, that is just a statement that (in my opinion) is senseless. > > > > Only in the context of the Dedekind definition of infinity, which one is > > not obligated to consume wholesale. Until one can prove that transfinite > > set theory is "correct", no one is obligated to accept the theory at all. > > 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. > > 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. 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. Smiles, Tony
From: Tony Orlow on 25 Aug 2006 14:13 Dik T. Winter wrote: > In article <44eef750(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > Dik T. Winter wrote: > ... > > > Indeed. Not step by step. And so? I have said this before. I am quite > > > sure others have said that before. You can not get the set of naturals > > > by adding the one by one. It is the axiom of infinity that asserts that > > > that set does exist. > > > > And that it's infinite. > > No. The axiom of infinity only asserts that the set exists. That it is > infinite (if it does exist) is a theorem that can be proven. > > > 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. > > > > But the last is irrelevant in the context of the paragraph above. There > > > is no proof whether the reals can be linearly ordered or not. Unless > > > you assume the axiom of choice. And under other discussions you can > > > prove that the reals can not be linearly ordered (I think). So *of > > > course* Cantors proof is not a proof that the reals cannot be linearly > > > ordered. > > > > If Ross and I each have our linearizations of the reals, then they > > exist, independent of the axiom of choice (which might as well be called > > the axiom of free will). > > You both do not have an linearisation of the reals. You both are doing > other things. Describe those "other things". How are they "other"? > > > > > > No. Nowhere in his papers did Cantor use decimals. > > > > > > > > Perhaps it is just the explanations of his second proof of > > > > uncountability which I have seen. He did use SOME digital representation > > > > of his list of reals. Was it binary? > > > > > > None at all. The second proof (the diagonal proof) is *not* about reals. > > > > 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. > > > > > > You make here as much sense as in much earlier articles. > > > > > > > > What part of does not make sense? I suppose it's hard to put that > > > > concept into a sentence or two and be readily understood. > > > > > > It works for finite sequences, but your conclusions do *not* work in the > > > infinite, because transfinite induction fails. > > > > I'm not using "transfinite" induction, but straight-up infinite > > variables and the notion that any infinite is larger than any finite. > > Quite simple stuff. > > You are trying to talk about standard mathematics, but not doing standard > mathematics, but some highly non-standard mathematics. It may work for > infinite sequences in your non-standard form of mathematics, but not in > common standard mathematics. > > > > > > > I don't disagree that the first is infinite while the second is > > > > > > unbounded bt finite, and therefore smaller. > > > > > > > > > > Pray, for once, provide a definition. A set is either finite or > > > > > infinite. And if a set is finite, by the definitions there is a > > > > > largest element. So, how do you define "unbounded but finite"? > > > > > > > > Not with the Dedekind definition. A truly infinite set must have > > > > elements infinitely many position beyond other elements, the way I > > > > see it. > > > > > > Yes, but in that case you are not using standard mathematical terminology. > > > And I have yet to see a *definition* of you about *infinite*. > > > > 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 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. > > > > > In the second proof of uncountability of the reals, there are no > > > > digital strings? What list is he deriving the antidiagonal from, > > > > a shopping list? > > > > > > Pray read. His second proof is *not* about the uncountability of the > > > reals. There are no reals involved at all. It is about (infinite) > > > sequences of symbols (he uses 'w' and 'm'). > > > > 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 are using words without definition, using words from standard > > > terminology but with (apparently) not the standard definition. That > > > is what makes you on occasion not understandable. > > > > 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? Tony
From: Virgil on 25 Aug 2006 14:13 In article <1156500736.055279.173430(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > Albrecht wrote: > > > We must conclude that we can only say: we can find infinite many points > > > on a line. This is a potential infinity. > > > > Do you have axioms for your mathematical theory of potential but not > > actual infinity? > > Five Peano axioms. Which version? In some versions, it is explicitly required that the set of all "naturals" exist.
From: Virgil on 25 Aug 2006 14:14
In article <1156500782.651084.171830(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1156148773.420187.122140(a)h48g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > You are dreaming. It does follow from the form of the numbers of my > > > list. > > > Try to come back to reality. Simply show how a digit position n can be > > > indexed the complete number of which, i.e., the complete sequence of > > > digit positions 1 to n is not in the list. > > > > The Hilbert Hotel method! > > Why does the Hilbert Hotel method not apply to my list, but only to > Dik's number? Because it DOES apply to your list! It applies to any list. |