An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Dik T. Winter on 2 Oct 2006 21:34 In article <1159649021.675137.307470(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > So, again, no definition. Where did I not speak the truth? > > > > > > Here: "...because he never answered to questions about it". > > > > You gave on Usenet a definition of natural number in answer to my question > > for it. I posted questions about that definition, and you never answered > > them. So in what way is it a lie when I state that you never answered > > them? > > I did not see an unanswered question. Apparently you did not see the questions at all. You stated as your definition something like: "numbers are in trichotomy with each other" and I answered something like "so omega is a number". But I never did see an answer. I asked for a mathematical definition of number. You never gave one (and also your paper does not give any mathematical definition), your only mathematical definition was about the trichotomy. > Please repeat. If you do repeat. > By the way, please switch to the thread "Cantor > Confusion" because this one has become too lengthy and, at home, I have > only a slow internet access. So I am not able to follow this thread > firmly. Oh. I did not know that slow internet access made long threads more difficult to follow than short threads. I see no reason to shift the subject. And certainly not to a subject for which a thread already does exist. > > > Most > > > questions on the representation of a number are answered in my paper. > > > > The questions were about the definition you gave in Usenet. And I do not > > ask about "representations", I ask for a *definition* of the concept > > "number". A proper, mathematical, definition. > > In my paper which you read I quoted Peano and v. Neumann on the first > page. Then, on page 2, I wrote "Of course this realization of 2 > presupposes some a priori knowledge about 2. But here we are concerned > with the mere realization." That is my point! A point, perhaps. But not a definition. > > > The definition of an object does not provide its existence. > > > > Indeed. But when it is *defined* as the limit of a sequence, and if that > > limit exists, that means that the object does exist. > > The limit omega does not exist. Who insists that the limit is omega, except you? 0.111..., seen as a decimal number exists (due to the definition) and is equal to 1/9. Seen as a termary number it also exists, and is equal to 1/2. Seen as a unary number it does not exist as a natural number. But in your sequence I see it only as a sequence of digits, without interpretation, and as such it does exist. > > > The cardinality of the set of prime numbers known today P(t) is as > > > unbounded as the time variable t of today. > > > > If you talk like that you can not talk about "the set of known prime > > numbers", because that is not a set, but a function of time. > > A set can be a function of time. No. A set is fixed. A function of time delivering sets is just that, a function of time delivering sets. In the same way sin(x) is not a real, it is a function of x delivering a real. > > > For > > each 't', the outcome is a specific set. A properly defined set does > > not change over time. > > That is your definition, not mine. We have the same with numbers. Of > course, every number is a constant. Nevertheless, variables can exist > for numbers (which can, but need not, be defined as sets) and for sets. Variables do exist, but are not numbers. > "The letters X and Y in these expressions are variables; they stand > for (denote) unspecified, arbitrary sets." Karel Hrbacek and Thomas > Jech: "Introduction to set theory" Marcel Dekker Inc., New York, 1984, > 2nd edition., p. 4. Yes, and so what? > > So when you talked about the set of known prime > > numbers, I thought you were talking about the set of prime numbers known > > at the time you wrote it, as I can give it no other interpretation. > > You fall back behind Cantor. He could. "ein in Ver?nderung Begriffenes > Endliches, das also in jedem seiner actuellen Zust?nde eine endliche > Gr??e hat." An example is the largest natural number you can > imagine. Try it. There is always a larger one. What is the point? Graham's number is, eh, quite large. And there are larger numbers. So, again, what is the point? -- 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 2 Oct 2006 22:16 In article <1159726829.184763.303470(a)i3g2000cwc.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > stephen(a)nomail.com schreef: > > > I suppose I should clarify this. You can approach the infinite > > using the the limit concept, but you always have to be careful > > when using limits, and you have to be precise about what you > > mean by the limit. > > Okay. But the point is whether there exist infinities that can _not_ > be approached using the limit concept. Yes. In ordinals they are called the "limit ordinals". The name is not very appropriate, I think, because they are the ordinals that defy the limit concept, but that is just naming. > Obviously they exist, because > how can we approach e.g. the Continuum Hypothesis by employing limts? I have no idea about the meaning of this statement, but off-hand I ould say that there is no way. -- 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 2 Oct 2006 22:12 In article <1159711218.812268.276490(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1159648393.632462.253170(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: .... > > > There is a bijection *possible*, but that does not mean that this > > > bijection is ruling the set number of numbers of sets. > > > > "set number of numbers of sets"? What does *that* mean? > > Meant is "numbers of elements2, alas everything is a set. "set number of numbers of elements" still makes not sense. Care to explain? > > Your error is still that you believe that a cardinal number should act > > as a natural number. You have some presupposed ingrated definition for > > the word number that I do not understand. Would you be happy if we used > > the terms "ordinal fluffs" and "cardinal fluffs". Bijection rules the > > "cardinal fluffs". Are you happy now? > > > Here I am interested in the possible indexes and the number of 1's in > 0,111... . What else you my do, derive define, and prove with your > fluffs interests me less. So you are only concerned about the naming, not about the concepts. > > > (There are exactly twice so much > > > natural numbers than even natural numbers.) > > > > By what definitions? You never state definitions. > > By the only meaningful and consistent definition: A n eps |N : > |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. > Do you challenge its truth? No, I never did. But you draw conclusions about it about the set N. Indeed, for each finite n, it is true. But this is *not* a proper definition for the amounts involved in infinite sets. Given two infinite sets A and B, by what method do you determine whether A has more elements than B, or the other way around? Are there more Gaussian integers than Eisenberg integers, and if so why? And if not, why not? > > Well, I almost always gave in this context the word number in quotes. > > So your assumption that I assumed a uniquely defined number 0.111... > > is wrong. Read back some time ago where I said (explicitly) that the > > only way we could interprete it was a string of digits. Note especially > > the article were I wrote about three possible interpretations. In my > > latest articles I simply interprete it as a string of symbols, that > > you wish to see it otherwise is your responsibility. > > Your main argument and my main target is the full presence of all > digits and their indexibility by numbers all of which do not cover > 0.111... . You attack the existence of an infinite sequence, and so also the existence of an infinite set. So be it. But that is just a negation of the axiom of infinity. With that axioms such things do exist. > > But it is the representation of some numbers, but that entirely depends > > on how you interprete that string of symbols. > > "..." is the representation of potential infinity, nothing more and > nothing less. 0.111... cannot be interpreted as you try to do. It is nothing at all until a definition has been given. Within mathematics there is no meaning without definition. > > > You could come up with that argument for arbitrary numbers, but not for > > > unary numbers. what you require is impossible. Either 0.111... is > > > larger than any number of the list, then you have to give a position > > > which is not covered by a list number or not. > > > > Now you are interpreting it as *number* with some additional connotations. > > I use your language if I discuss with you. I would never use such names > like aleph in a monolog. Apparently you do not understand my language, which is mostly the language of mathematicians. In that case discussion is fruitless. > > Well, yes, if you wish to interprete it as number, I would say it is > > omega in unary notation. And so it is larger than any number on the list. > > What you mean with that I should give it a position is unclear to me. > > I mean: Indexing of all digit position of 0.111... by the unary numbers > 0.1, 0.11, 0.111, ... is impossible unless all digi positions f > 0.111... are also covered by these unary numbers. Again, confused. Each digit position can be covered, but there is no entry in the list that covers all digit positions. On the other hand, every digit can be indexed (when we interprete the elements of the list as unary natural numbers). > Instead of "to index position" we can also say "to cover up to position > n". Hence you assert that it is possible to cover 0.111... up to every > position but it is impossible to cover every position. Yes. > > > Take into account that also Cantor's diagonal argument cannot be > > > executed in unary representation. > > > > Two red herrings in a single sentence. Can you get more? > > (1) Cantor's diagonal argument was about countable sequences of two > > symbols. There is only one countable sequence of one symbol. > > Cantor's argument was about reals. He strived for generality but did > not see that two symbols are not enough. You are seriously wrong. > > (2) Cantor's argument as augmented by Zorn and later by somebody who > > I do not know can not be executed for reals represented in base > > 3 or smaller. But reals are not tied to their representation. > > Therefore a general truth should not depend on the base 4 or larger. The *proof* (as modified by someone unknown to a proof about reals) depends on base 4 or larger. This does *not* mean that the truth depends on it. When I give a proof about reals in whatever base notation I wish, the truth remains a truth when I change base, although the proof may possibly not be valid in that base. If I give a proof that in base two all primes, except one, have a last digit 1, and so are odd, that does not mean that that proof is valid in other bases, but the result remains valid (all primes, except one, are odd). > > > Of course you can set up a bijection beween the sets > > > k + omega = {-k, -k+1, -k+2, ... , 0, 1,2,3,...,} and > > > -k + omega = {k+1, k+2, k+3, ...}. > > > But that does not mean that both sets have the same number of elements. > > > > You are utterly confused. Ordinals are concerned with order preserving >
From: cbrown on 2 Oct 2006 23:49 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> Tony Orlow wrote: > >>>> cbrown(a)cbrownsystems.com wrote: > >>>>> Tony Orlow wrote: > >>> <snip> > >>> > >>>>>> Like I said, there were > >>>>>> terms in my infinitesimal sections of moving staircase which differed by > >>>>>> a sub-infinitesimal from those in the original staircase. So, they could > >>>>>> be considered to be two infinitesimally different objects in the limit. > >>>>> Here's a thing that confuses me about your use of the term "limit". > >>>>> > >>>>> In the usual sense of the term, every subsequence of a sequence that > >>>>> has as its limit say, X, /also/ has a limit of X. > >>>>> > >>>>> For example, the sequence (1, 1/2, 1/2, 1/3, ..., 1/n, ...) usually is > >>>>> considered to have a limit of 0. And the subsequence (1/2, 1/4, 1/6, > >>>>> ..., 1/(2*n), ...) which is a subsequence of the former sequence has > >>>>> the same limit, 0. > >>>>> > >>>>> But the way you seem to evaluate a limit, the sequence of staircases > >>>>> with step lengths (1, 1/2, 1/3, ..., 1/n, ...) is a staircase with > >>>>> steps size 1/B, where B is unit infinity; but the sequence of > >>>>> staircases with step lengths (1/2, 1/4, 1/6, ..., 1/(2*n), ...), which > >>>>> is a subsequence of the first sequence, would seem to have as its limit > >>>>> a staircase with steps of size 1/(2*B). > >>>>> > >>>>> Unless steps of size 1/B are the same as steps of size 1/(2*B), I don't > >>>>> see how that can be possible. > >>>>> > >>>>> Cheers - Chas > >>>>> > >>>> It's possible because no distinction is currently made between countable > >>>> infinities, even to the point where a set dense in the reals like the > >>>> rationals is considered equal to a set sparse in the reals like the > >>>> naturals. Where there is no parametric understanding of infinity, > >>>> infinity is just infinity, and 0 is just 0. > >>> Uh, OK. I assume that you somehow resolve this lack of "parametric > >>> understanding" in /your/ interpretation of T-numbers. > >> Well, yes. That's the whole point, but I'm not sure which particular > >> numbers you mean. Maybe the T-riffics, which are center-indefinite > >> digital numbers? > >> > > > > I mean, whatever type of number the /particular/ number "x" is; when > > you state "/the/ limit of the staircases has steplength 1/x". > > > > If you say you have some particular infinite number of steps x... I /don't/ say that; you do. What do you mean by it? >, then you > can consider 1/x to be the specific infinitesimal size of each. As long > as 1/x is considered nonzero Well, do you, or do you not, consider it non-zero? >, there is a ratio between the x and y > offsets of any given segment which gives a direction at that point. It > is clear that when defined this way the directions of the segments in > the staircase do not at all approach the directions of the corresponding > segments in the diagonal, even the locations of the endpoints of the > segments do become arbitrarily close. But they don't become arbitrarily close. For example, since 1/x^2 < 1/x in your system, and yet your points do not get closer than 1/x, there is still "more room" for them to get closer - they /don't/ get arbitrarily close.. > > >>>> Where there is a formulaic > >>>> comparison of infinite sets as n->oo, the distinction can be made. The > >>>> fact that you have steps of size 1/n as opposed to steps of size 1/(2*n) > >>>> is a reflection of the fact that the first set has twice the density on > >>>> the real line as the first. As a proper superset, it SHOULD be larger. > >>>> So, it's quite possible to make sense of my position, with a modicum of > >>>> effort. > >>> Well, let me ask you this: > >>> > >>> Suppose we have the original sequence of staircases, with step lengths > >>> (1, 1/2, 1/3, ..., 1/n, ...). Let S be the T-limit staircase; you claim > >>> that it has step sizes 1/B, where B is unit infinity. > >> Well, where B is some infinite number of iterations for both staircase > >> and diagonal. > > > > "Some" number? Do you mean that there is no "the limit" of the > > staircases, but instead many possible "a limit" to the staircases? > > No, the limit is the limit. Once you have applied an infinite number of > steps you have become infinitesimally close to the limit, and have > essentially reached it. "Essentially" reached it? How is that different from "reaching" it? > Any infinite value will do. In the limit, as a > segment sequence, the infinite staircase continues to have a length of > 2/x for each step, and x steps, for a length of 2. > But if x = B, then the steps have length 1/B. If x = 1/B^2, then the steps have a /different length/; so these two "limits" can't be the same. > > > >> We can call it the unit infinity if you want, especially > >> since it covers a space of 1 unit. :) > >> > > > > One might as well start somewhere. > > > > Starting from 0 and 1, we can construct the naturals. Add the concept > > of "subtraction", and (arguably) we get the intgeres. Add the concept > > of multiplication and division, and we get the rationlas. > > Throw in exponentiation, logs and roots, and we get the reals. If you include roots of polynoimals, you get the algebraic closure of Q (normally called the algebraic numbers or A), which does not include transcendental numbers such as pi, e, and so on. In order to get the transcendentals, including "exponentiation and logs", you need a concept of limit; and your limit concept is currently too weak to allow this. > I suppose > I should get back to my H-riffic numbers. > Let's stick to your assertions that you already claim are true, such as that the limit of the staircases has length 1/x. > > > > Starting from the reals and B, one can talk about B+1, B-1, 1/(B-1), > > etc. These may not perhaps cover /every/ infinite "number", but from > > your previous statements, certainly for /some/ B these "numbers" exist. > > Well, I think by declaring such a unit infinity as the number of reals > in the unit interval has its applications. Add to that the application > of induction to such numbers, as if they really exist on the same > continuum with the finite reals, and you have a rich means of > distinguishing infinite sets. I guess my approach seems too "normal" for > infinite numbe
From: imaginatorium on 3 Oct 2006 00:28
Lester Zick wrote: > On Mon, 2 Oct 2006 16:43:15 +0000 (UTC), stephen(a)nomail.com wrote: > > >Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >> stephen(a)nomail.com wrote: > > [. . .] > > >So one wonders what criteria you used to determine that > >this infinity cannot be approached via limits. > > Which kinda takes us back to the definition of infinity wouldn't you > say? If infinity can be approached via limits infinity would have to > refer to the number of infinitesimals and if not infinity couldn't be > approached via limits. Oh Lester, more poetry. ... pure poetry. How will you respond to this post, I wonder? Brian Chandler http://imaginatorium.org |