An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Dik T. Winter on 25 Aug 2006 11:21 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 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. > > > > 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. > > > > 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"? 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? > > > 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. > > 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. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: MoeBlee on 25 Aug 2006 12:20 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. In what logic? I have to think that it is at least second order logic, since five Peano axioms in first order logic isn't even as strong as the full seven Peano axioms in first order logic. So what is your logic (and I don't suppose it's classical?), and which version of the Peano axioms do you have in mind? (I'm guessing it's second order: (1) N0, (2) Nx -> NSx, (3) Sx = Sy -> x=y, (4) ~Ex 0=Sx, (5) Induction schema.) Then, in this system, what are your definitions of 'potential infinity' and 'actual infinity'? MoeBlee
From: Tony Orlow on 25 Aug 2006 12:29 David R Tribble wrote: > Dik T. Winter wrote: >>> On the other hand, do not confuse 0 with the infinitesimals. And there >>> is no aversion to the infinitesimals. Look at the surreals, look at >>> non-standard analysis. Infinitesimals abound. Still, in order to be >>> logically consistent, 0 has a special role. > > Tony Orlow wrote: >> Yes, true. Absolute 0, the origin, is a single point, and >> infinitesimals, neighboring points, are something ever so slightly >> different. But then again, oo as a concept and a limit is something >> absolute and different from, say, the number of points per unit of >> space, or some other infinite unit. To have infinitesimal numbers, you >> need to have non-absolute infinities also. > > That's not correct. To have infinitesimals, where > for infinitesimal e, 0 < e < x for all real x, > it is sufficient to define them as reciprocals of "illimited" numbers, > where > for infinitesimal e, e = 1/h, where h is illimited; > for illimited h, x < h for all real x. > Okay, but how is that different from what I said. Specific infinities, or illimited numbers, go hand in hand with the notion of infinitesimals. > These illimited numbers may have properties like the reals (e.g., > order, addition, multiplication, etc.), but are not reals because they > exist outside R. But these illimited (or "unreal") numbers do not > have be "infinite" numbers - they just have to be numbers with > magnitudes larger than any real. (Obviously, these numbers do > not exist in standard arithmetic.) No, but they are more compatible with standard finite mathematics than the transfinite variety of infinity. Do you consider them compatible with limit ordinals and infinite cardinalities? > > Indeed, the non-real numbers in non-standard analysis are often > called "illimited" rather than "infinite". One reason is that they do > not act like infinite numbers (infinite ordinals or cardinals), so it's > misleading to call them "infinite". > Hmmm. On the other hand, since they behave like other numbers, it's fitting to call them numbers in the first place, which is not clearly the case with limit ordinals and transfinite cardinalities. I would imagine they were called "illimited" so as not to step on Cantorian toes, since transfinitology already claimed the first definition of "infinite". Tony
From: Tony Orlow on 25 Aug 2006 12:38 Virgil wrote: > In article <44ed99b7(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Dik T. Winter wrote: >>> In article <1156363640.845840.187460(a)75g2000cwc.googlegroups.com> >>> mueckenh(a)rz.fh-augsburg.de writes: >>> > >>> > Dik T. Winter schrieb: >>> > >>> > > In article <ec8if0$rtd$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu >>> > > writes: >>> > >>> > > > Indeed, I agree with WM's logic concerning the identity >>> > > > relationship >>> > > > between element count and value in the naturals. He's quite correct >>> > > > in >>> > > > that regard. >>> > > >>> > > Well, you and he are not. The logic is flawed. >>> > >>> > For 1, 2, and 3, order and value are identical, even according to your >>> > logic, I suppose. Where does the first deviation happen? Which one does >>> > deviate first, i.e., which one does first be larger than the other? And >>> > why? >>> >>> It is the case when the set size is not a natural number. >> And, with the addition of which natural number to the set does the set >> achieve this unnatural size? > > TO is knowingly presuming something contrary to fact. > > The axiom of infinity guarantees a set whose 'size' is not that of any > natural in any system which contains it. > >>> 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. > > TO has claimed it, but not proved it. > TO claims a lot but never proves any of it. > Since the set of digit positions in any positional notation for the > members of N is indexed by N, the cardinality of the set of bit > positions equals the cardinality of its index set. Since the bit strings are the power set of the set of bit positions, each natural being a unique subset of 1 positions, the set of naturals is power set to the set of bit positions. In your concoction, the naturals are power set to nothing. Rather a great blunder to have go so long unnoticed. > > Note: Note that for any n in N, the least natural requiring n bits is > 2^(n-1), but this number of bits suffices for all naturals less than 2^n. True. So? > > As the density of values of form 2^(n-1) approaches zero as n increases > without bound, so does the probability that the number of bits required > does not allow more than is required. Yes, the chances that some randomly selected infinite number of bit strings can be exactly represented by some infinite number of bit positions is infinitesimal. > >> Yes, given the current understanding, any unbounded set is infinite. > > Find any dictionary anywhere that says otherwise, that allows TO's > self-contradictory unbounded but finite. Dictionaries reflect the widest and most accepted usage of words. >>> A set is infinite if it is not finite. A set is Dedekind infinite if >>> it can be mapped to a proper subset of itself, and it is Dedekind >>> finite if it is not Dedekind infinite. When we assume the axiom of >>> choice, the two notions are identical. Without that axiom there >>> can be infinite sets that are Dedekind finite. Do you want to know >>> more about set theory? >>> >>> Now, using, this terminology (pretty standard), what is a "finite but >>> unbounded" set? >> Using that system, the phrase is senseless, but that system is not the >> real universe. It's a concoction. > > TO's concoctions are even less sensible and less part of any "real" > universe. > > In every standard dictionary, finite means being bounded or having ends, > and endless means infinite. > > TO thinks he can get away with saying endless means having ends, but TO > is, as usual, wrong. When you have the set of reals in [0,1] there are distinct endpoints to the set. The question when dealing with an infinite set is whether enumeration of the elements therein can ever terminate as a process. Where there is a distinct range and the enumeration never completes, the set is infinite. Unfortunately, because the enumeration of the naturals never ends, that set is considered actually infinite, when it's really only potentially so. The endlessness is due, not to actual infinitude, but to the non-boundary between finite and infinite. :)
From: MoeBlee on 25 Aug 2006 12:39
Tony Orlow wrote: > If the naturals are not a subset of the reals in ZFC and NBG, then those > theories are even more screwed up than they already seemed. With either of the usual constructions of the reals (Dedekind cuts or equivalence classes of Cauchy sequences), both the system of naturals and the system of rationals are isomorphically embedded in the system of reals. Mathematicians usually speak of the system of natural numbers and the system of rational numbers as subsystems of the the system of reals. Strictly speaking, that is incorrect, but it is harmless given the isomorphism. > >>> I don't know what is meant by '(y=2)' in the larger formula. > >> I mean that for x>2, 2*x < x^2 < 2^2. For x=2 they're equal. > > > > Then (1) is a very peculiar way of trying to convey that claim. > > What's peculiar about it? The notation of appending the formula with the parenthetical is unclear without your explanation of its meaning. This is a minor point only regarding your notation. > >> Is aleph_0>2? > > > > AS TO's comparison operator is according to TO defined only for reals, > > the question is nonsense. > > > > Then aleph_0 is not a count of any sort, or it would exist on the > hyperreal line. It's not a number of any real sort. 'number of any real sort' is not a defined predicate of set theory. w is not in the standard ordering of the reals. That does not make w an undefined object. > you don't see scientists accepting transfinitology either. Of course, you have a survey of scientists to support your claim. > > As we have no definition of length valid for intervals whose endpoints > > are not finite reals, it means nothing. > > > > That's because you have no infinite values which behave sufficiently > like numbers to support such a definition. You have no coherent system of definitions at all. It's not the job of set theory to define objects that obey the whims of your informal notions. > The selection of any unit is done by simply choosing a point separate > from the origin. When it comes to division using infinite values, one > translates the geometric definition into symbolic form and applies > induction formulaically. "Translates the geometric definition into symolic form and applies induction formulaically" is no less doubletalk than "Coordinates the numeric form into geometric postulates and applies the recursive definition metrically." MoeBlee |