An uncountable countable set
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From: Virgil on 24 Aug 2006 15:14 In article <44ed987e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David R Tribble wrote: > > Tony Orlow wrote: > >>> Because it's a set of consecutive naturals starting at 1. > > > > Virgil wrote: > >>> It doesn't matter where it starts, the issue is whether it ends with a > >>> "largest natural". In standard mathematics it does not. > >>> TO seems to switch postions erratically on the issue. > > > > Tony Orlow wrote: > >> Let's put it this way. The max and the size of the set are equal. If one > >> exists, the other exists as well, since it's the same. If one does not > >> exist, then neither does the other. > > > > A more logical conclusion would be that if a set has no largest member, > > then its size is not equal to any finite number. Which happens to > > be the case. > > > > Just because a set doe not have a largest (or smallest) member > > does not mean it mysteriously does not have a cardinality. Logic > > dictates that every set must have a cardinality, either zero, a > > finite cardinality, or an infinite cardinality. > > > > "Logic dictates"? Is this a theorem? It is a theorem in ZFC and NBG that every set has a cardinality, It is not a theorem in ZF, since absent an axiom of choice, or equivalent, there is no way of guaranteeing that functions between arbitrary sets even exist. However for ordinals, it is a theorem even in ZF that the all have cardinals. > If so, it is contradicted by the > non-existence of a cardinality for the set of bit positions required to > list the naturals in binary. A countably infinite set of bit positions is sufficient. No finite set of bit positions is sufficient. Therefore the cardinality in question is that of N, and exists quite nicely than you. If TO objects to this, let him state the cardinality of the set of binary bits necessary to express every natural up to, say, six, but yet not able to express any larger natural. >That can be neither finite nor infinite, > without producing a contradiction. The only contradiction is in TO's assumption that what is just barely sufficient for N cannot be sufficient for other things. > > > > >> I have never said I think there is a largest natural. I have said that > >> some of your assumptions lead to that conclusion. > > > > You assume that the axioms of set theory lead to that conclusion, > > but you've never proved it. > > > I don't assume it. Your assumption of a smallest infinite leads to that > conclusion. Logic dictates it. Let's see that alleged logic. Let's see TO's *proof*, based only on the axioms of, say, ZF, that there is a largest natural. In every one of TO's arguments to that effect so far presented, he has snuck in at least one of his own assumptions which is no part of any standard set theory. If TO thinks he can do it now without any such cheating, let him try!
From: David R Tribble on 24 Aug 2006 15:34 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. 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.) 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".
From: Virgil on 24 Aug 2006 15:37 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. 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. 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, 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. > > > 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.
From: David R Tribble on 24 Aug 2006 15:43 Albrecht Storz wrote: >> Infinite sets are self contradicting. > Virgil schrieb: >> Not in ZF or NBG. What are the axioms of Storz's system? > Albrecht Storz wrote: > There is no relevance in which system the axiom is found. > E.g. the Axiom A: "Axiom A is wrong", is self contradicting, regardless > of which other axioms are used, I think. The same holds for the axiom > of infinity. How is the Axiom of Infinity self-contradictory?
From: David R Tribble on 24 Aug 2006 15:44
Tony Orlow wrote: >> Please >> show where I am being inconsistent, not with set theory, but within my >> own assumptions. Remember, I don't claim to believe in transfinite set >> theory, and don't intend to be consistent with it. > David R Tribble wrote: >> The problem is that your own assumptions are not consistent with >> each other - they don't form a coherent theory. Many of these >> inconsistencies have been pointed out to you before, but you choose >> to not believe them or simply to ignore them. > Tony Orlow wrote: > It's easy to say that without mentioning any specific inconsistencies. > > In fact, mapping the naturals in [1,Big'un] to the reals in [Lil'un,1] > using the mapping function f(x)=x/Big'un yields Ross' Finlayson numbers, > and is perfectly consistent with IFR. Not only do we obviously have > Big'un^2 reals on the line because we have the sum of Big'un unit > intervals each containing Big'un reals, ... That implies that there are BigUn naturals in the real number line, so that |N| = |R|. You state this as though it's a fact, but what is your proof? |