An uncountable countable set
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From: David R Tribble on 23 Aug 2006 19:03 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. > 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.
From: Virgil on 23 Aug 2006 19:10 In article <ecih5d$33a$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > < snippety doo dah > > > >>> Which is sufficient justification for refusing to allow TO's extension > >>> of induction beyond standard induction. > >> Or, alternatively, sufficient justification for rejecting > >> transfinitology and the notion of a specific boundary between finite and > >> infinite. > > > > TO is the one who demands a "specific boundary" between finite and > > infinite, what we rerquire and what we have is specific definitions of > > what constitutes a finite set and what constitutes an infinite set. > > No, what my system requires is a workable variable that's consistent > whether it's finite or infinite. What "system" is that, TO? When TO presents his as yet mythical system, only then can he truthfully speak of having a system. > > >>>>> The only acceptable forms of induction in ZF or NBG are finite > >>>>> induction and transfinite induction, neither of which has TO shown he > >>>>> knows how to apply. > >>>> But I am explicitly suggesting the alternative, that if f(x)>g(x) for > >>>> all x>y, then it is true for all infinite x. This leads to a clearer set > >>>> of conclusions. > >>> TO's alternative also leads to inconsistencies with the status quo ante, > >>> which is sufficient reason for rejecting it. > >> Huh? You mean, if I reject your axiom system, and concoct another, the > >> fact that yours came first means that contradictions between the two are > >> the fault of mine? > > > > What I mean is that you cannot impose your axioms/assumptions on top of > > our ZF, ZFC of NBG and then object to the contradictions which arise as > > being solely due to our axioms. > > I don't. I freely adimt regularly that the contradictions arising from > my assumptions are between those and the standard assumptions. Others > claim all contradictions arise from within my assumptions, which isn't > the case. Until TO has a complete system presented for examination by others, he is not in a position to state that he even has a system, much less that it is free of inconsistencies. > > > > > Our axioms sets do not allow your form of induction. Assuming your > > version in addition to ours gives rise to contradictions, so we reject > > your assumptions as contradictory. > > You're absolutely correct. I do not want you to consider infinite-case > induction in ADDITION to ZFC or NBG, but as an ALTERNATIVE. They are > clearly mutually incompatible. We are sufficiently satisfied with ZF and NBG, and find nothing in TO's alternatives to excite anything but our revulsion, so why should any of us want TOmania? > > > > > If ever TO completes his own system of axioms, including if he wishes > > his version of induction, we shall see whether it has obvious > > inconsistencies. Ours don't. > > > > Okay, well, when the baby's back in daycare (1 week) and the boys back > in school (2 weeks) then I'll have more time, but this is good prep > work. Thanks. :) I see. TO's eternal "tomorrow" > > Having a system does count, and while we have several, TO has none. > > I have one, but not polished and published yet. Still, it's enough for > you to stop objecting out of pure habit. When I actually see TO's system, I will stop asking to see it. > > > > >>> As TO has shown no talent for recognizing truth, and even seems to have > >>> a talent for avoiding it, his judgements on where mathematical "truth" > >>> lies are, at best, untrustworthy. > >>> > >> Define "truth". > > > > The tautologies of formal logic are truths. Those are the only ones > > mathematicians qua mathematicians can be sure of. > > Anything else is uncertain. > > Then the axioms of ZFC and NGB are not "true". So, stop pretending they > are. There are other valid perspectives on the matter. > > > > >>>> No, transfinitology doesn't satisfy my spiritual needs. > >>> For spiritual needs, one needs something a little less literal minded > >>> than mathematics. > >> What makes you think mathematics is literal? It's certainly not > >> concrete, but very abstract. What the numbers refer to is anyone's guess > >> when they're doing the algebra. That's the beauty. It only becomes > >> literal upon application. > > > > Absolutely backwards! Mathematics is at its most literal minded when > > most abstract. To the extent that it becomes applied it loses its > > literal mindedness. > > Define "literal". I read it as "actual". > > >>>>> What does one do for sets which are not ordered, or ordered sets which > >>>>> have no max?, both of which are abundant. > >>> > >>>> Examples? Symbolic sets (languages) need not be ordered to be measured, > >>>> necessarily, and sets which have no distinct range have no distinct > >>>> size, unless the elements themselves have no relative measure. > >>> The multidimensional point sets of multidimnsonal spaces, such as real > >>> vector spaces, are not ordered and are not orderable in any way > >>> consistent with their geometric properties. > >> . It's not linear > > > > Thus, being non-linear, it does not fit on any number line. > > > > By which TO admits his error. > > Uh, no. With the snip of context, that doesn't even make enough sense to > respond to. > > TO
From: Dik T. Winter on 23 Aug 2006 19:09 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. > The cardinal number aleph_0 is infinite while the ordinal number > remains finite in order to have infinitely many finite numbers. This makes absolutely no sense. The set of all natural numbers (all finite) has cardinal number aleph-0 and the ordinal number of that set is w. > > And his proof is not a proof. > > O course not, because every proof which has an unpleasant result is not > a proof. Then show that the set of all natural numbers does not have cardinal number aleph-0 and ordinal number w. A proof please. > > > For my part, I agree that the set of finite > > > naturals is finite, though unbounded, > > > > In that case you are not using standard mathematical terminology. I > > have no idea what a finite but unbounded set is. > > That's why you cannot understand mathematics. You fall back behind > Cantor. He knew it. Oh, perhaps *you* do not understand mathematics? Earlier I have already written that a few things that Cantor has written do not conform with current thinking. But see <http://mathworld.wolfram.com/FiniteSet.html>: A set X whose elements can be numbered from 1 to n, for some positive integer n. 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? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Tony Orlow on 23 Aug 2006 19:17 MoeBlee wrote: > Tony Orlow wrote: >> Set theory contradicts with: >> >> (1) E y e N, A x>y, x< 2*x < x^2 < 2^x (y=2) >> >> because: >> >> (2) A y e N, aleph_0>y > > I don't know what you intend '<' to stand for. For the domination > relation? The less than relation on ordinals? The standard "less than" operator, commonly used for finite reals. > > 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. Is aleph_0>2? > >> and >> >> (3) aleph_0/2 = aleph_0 = aleph_0^2 < 2^aleph_0 >> >> (1) is trivially inductively provable. > > Do you mean (1) is a theorem of set theory, or do you mean it is > provable that (1) is the negation of a theorem of set theory? That (1) contradicts set theory. It's certainly not a standard theorem, in the infinite case, but it should be. However, its incompatible with aleph_0 and the system of limit ordinals. > >> (2)and (3) are from transfinitology. > > What is transfinitology? You know, Cantorian religion. That Hollywood stuff. ;) > What is the definition (and in what theory is > this definition?) of '/' where w (omega) is in the numerator? Well, I was calling it Bigulosity. The definition of x/y is "how many intervals of length y fit into an interval of length x?". This is constructible using straightedge and compass. Where it's not integral, is as approximable as it is with symbolic reals. > > MoeBlee > Have a nice day! Tony
From: Dik T. Winter on 23 Aug 2006 19:31
In article <1156363768.777975.223810(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > It is the same with the staircase: If the total height is H, then there > > > must be at least one stair of height H. > > > > Wrong. Think asymptote. > > Now I understand your idea and your error. You intermingle the height > of the staircase and its least upper bound, which is not the same. As > long as you cannot find a stair that has height 1 you cannot assert > that the staircase had the height 1. But in that case the same holds for the width. > It even if you argue that no > smaller quantity can be named as height of the staircase, then I must > tell you that this fact is solely due to the decreasing differences of > the stairs, converging to zero. And similar holds for the width. So either you state that nor the height, nor the width are 1, or you state that both are 1. That depends entirely on your viewpoint. But as the height and width of my staircase at step n is 1 - 1/2^n, I would say that the height and width of the completed staircase is the limit of that number. Apparently you disagree. > There is no asymptote. Hence your > comparison of 1 and oo fails. Yes, I wanted to simplify the picture to show that what holds for the height also holds for the width. > > > > All digit positions are indeed in the list, and it is itself not in > > > > the list. Why must it be in the list itself? Consider the digit > > > > positions as bricks. I state the number I have is built of bricks, > > > > but I do not state that the number I have is a brick itelf. That > > > > is what *you* are claiming. > > > > > > You forget that the numbers of my list leave no other outcome. > > > > But they do. > > No. Not the stairs with constant height difference. This section was not about that. > > > We have infinitely many differences between natural numbers. Hence they > > > sum up to a non-natural number. Contradiction. > > > > What is the contradiction? There is nothing in mathematics from which you > > can find that "an infinite sum" of natural numbers is a natural numbers. > > (And I put that in quotes because that is not really defined in > > mathematics.) > > As long as we are in the naturals we know: Each natural is the sum of > all preceding differences. Infinitely many differences require an > infinite sum. Wrong. In mathematics the concept of "infinite sum" is not defined, as I have told you already *many* times. We can talk losely and state that the infinite sum of all those numbers is indeed aleph-0, but note that that is talking losely. All infinite sums as in sum{n = 1 .. oo} used in mathematics are defined by limits. > If you insist that there are infinitely many naturals and > if infinity is a number aleph_0, then there is also a magnitude > aleph_0. Makes no sense, again. What do you mean with the term "magnitude"? The only connection I know of is where it is used in connection with real or complex numbers. And it can also denote some norm (in general Euclidean norm) in vector spaces. And as the finite cardinals equate to the non-negative integers, that equate to their magnitude, you might consider a cardinal number to represent its own magnitude. Do you mean that? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |