An uncountable countable set
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From: Tony Orlow on 1 Oct 2006 17:29 Virgil wrote: > In article <45201597(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <451df438(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <451dd293(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>>> Virgil wrote: >>>>>>> In article <451d83c4(a)news2.lightlink.com>, >>>>>>> So what balls remain in the vase at noon, oh waffler extraordinary? >>>>>> For n balls inserted, balls n/10+1 through n remain at the end of any >>>>>> iteration n. You specify the number of iterations, I'll give you the >>>>>> sum. What was it? Aleph_0? >>>>> All iterations executed before noon. >>>> And that would be how many? >>> All of them. >> How many in quantitative terms? All the people in the world doesn't >> describe the global population as a quantity. > > "Quantitative terms" is TO's bag. If he want that he will have to figure > it out for himself. "Al of them" is enough for me. Does you wife know about 'Al'? If you want to do math, you might want to stick to quantities. It's much easier. Tony
From: Virgil on 1 Oct 2006 17:31 In article <45202ef3(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > Hi Virgil. You look nice tonight. Did you get your hair done? Nice of you to notice what I have so little of. > Can I make the balls disappear or not? How long does it take to get > permission to speak from your superiors? What is YOUR judgment? > > > > If one is given a certain set of rules which produce a certain outcome > > then any alteration of those rules will produce an outcome irrelevant to > > the consequences from the original set of rules. > > If one changes a rule in a specific way, it will have a specific > consequence. But not necessarily relevant to what would occur without the change.
From: Virgil on 1 Oct 2006 17:49 In article <45203282(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <1159711218.812268.276490(a)c28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >> Dik T. Winter schrieb: > >> > >>> In article <1159648393.632462.253170(a)k70g2000cwa.googlegroups.com> > >>> mueckenh(a)rz.fh-augsburg.de writes: > > > >>> > (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? > > > > I challenge the "truth" of its being the ONLY meaningful and consistent > > definition. > > > > "Mueckenh"'s claim is like that of a blind man claiming that colors are > > imaginary. > > They are, even for the seeing. > Then how is it that spectrographs can detect them? > > > > > I, and many others, find both meaning and consistency in the definition > > of cardinality. That "Mueckenh" does not is more a measure of his > > incapacity than of any lack of meaning and consistency. > > WM, HdB, Finlayson, others and I see that the definition is lacking. > There's also Zuhair and Petry, and a slew more. This is the most > contentious of issues here. Perhaps you and I touched on the root of it, > The nature of logical implication. The thing is that hundreds or even thousands agree with me and your group do not even agree with each other. > > Can you appreciate out striving for more. Not when you are so ignorant of what is already there. > Are you so complacent in your position? No, but I am reasonably complacent that any errors in my position will be expressible with standard logic. > > A specific proof of a general truth can be based on whatever it is based > > on. > > To the detriment of its generality. Every proof is specific in some ways, but not necessarily in ways that limit the generality of the theorem they prove. > > > > > There are other proofs , including Cantor's first proof, which do not > > depend on any sort of representations of the reals. > > > > Cantor's first is an interesting proof of the uncountability of the > continuum, and I consider it valid. It demonstrates that the notion > that, for numerical strings a, b, and c in set S containing > representation of all r in R, ((aeS ^ ceS ^ a<c) -> Eb ^ beS ^ a<b<c) -> > (Es ^ seS ^ A neN length(s)>n) That is certainly nothing like the first proof that I was alluding to. http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof (quote) The theorem Suppose a set R is 1 linearly ordered, and 2 contains at least two members, and 3 is densely ordered, i.e., between any two members there is another, and 4 is complete, i.e., if it is partitioned into two nonempty sets A and B in such a way that every member of A is less than every member of B, then there is a boundary point c (in R), so that every point less than c is in A and every point greater than c is in B. Then R is not countable. The set of real numbers with its usual ordering is a typical example of such an ordered set. The set of rational numbers (which is countable) has properties 1-3 but does not have property 4. The proof The proof is by contradiction. It begins by assuming R is countable and thus that some sequence x1, x2, x3, ... has all of R as its range. Define two other sequences (an) and (bn) as follows: Pick a1 < b1 in R (possible because of property 2). Let an+1 be the first element in the sequence x which is between an and bn (possible because of property 3). Let bn+1 be the first element in the sequence x which is between an+1 and bn. The two monotone sequences a and b move toward each other. By the completeness of R, some point c must lie between them. (Define A to be the set of all elements in R that are smaller than some member of the sequence a, and let B be the complement of A; then every member of A is smaller than every member of B, and so property 4 yields the point c.) The claim is that c cannot be in the range of the sequence x, and that is the contradiction. If c were in the range, then we would have c = xi for some index i. But then, when that index was reached in the process of defining a and b, then c would have been added as the next member of one or the other of those two sequences, contrary to the assumption that it lies between their ranges. (\quote) > > > >>> > You have not *shown* that, but defined it, erroneously. But if you had > >>> > shown it, then 0.111... was in the list, which also would have been > >>> > wrong. > >>> > >>> Stated without proof at all. What is erroneous about my definition? > >>> Do you assert that definitions can be erroneous? If so, why? Do you > >>> think the definition > >>> Let a be the number such that a = 4 and a = 5 > >>> is erroneous? I think not. It is a proper definition, but there is just > >>> no 'a' that satisfies the definition. > >> It is erroneous, because you say let a *be* which is false, if a cannot > >> *be*. > > > > There is no such thing as an "erroneous" definition, except possibly in > > the sense of a grammatically incorrect one. A definition may lack any > > instantiation, such as a 4 sided triangle, but as a definition is valid. > > yada blabba flob. Sure Virgule. TO at the peak of his rational reasoning again, I see.
From: Virgil on 1 Oct 2006 17:52 In article <452032b9(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45201554(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <451df41c(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> In article <451dd1f2(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>> Virgil wrote: > >>>>>>>> Are you saying that aleph_0 naturals only require > >>>>>>>> ln(aleph_0+1)/ln(2) > >>>>>>>> bit positions? > >>>>>>> Not at all. I am talking about indvidual natural numbers as members > >>>>>>> of > >>>>>>> N, ,not N itself, which is not a member of N. > >>>>>> And also for every set of contiguous naturals starting at 0 EXCEPT for > >>>>>> N. Why EXCEPT for N? > >>>>>> > >>>>> For the same reason that a paper sack holding oranges is not an orange. > >>>>> > >>>> A set is a sack? It is nothing besides the elements it includes. > >>> A set is a container, and is not one of the objects that it contains. > >> It is nothing more or less than its contents. > > > > It is determined uniquely and entirely by its contents, as stated in the > > axiom of extentionality. > > So we agree. There is nothing besides the members. To say that it is completely determined by its members is not to say that it "is nothing besides its members". If it were nothing besides its members we cold not give it a name as a thing of its own.
From: Tony Orlow on 1 Oct 2006 17:54
Virgil wrote: > In article <4520236c(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David R Tribble wrote: >>> Tony Orlow wrote: >>>>> For the sake of this argument, we can talk about infinite reals, of >>>>> which infinite whole numbers are a subset. >>> David R Tribble wrote: >>>>> What are these "infinite reals" and "infinite whole numbers" that you >>>>> speak of so much? >>>>> >>>>> If you've got a set containing the finite naturals and the "infinite >>>>> naturals", how do you define it? N is the set containing 0 and all >>>>> of its successors, so what is your set? >>> Tony Orlow wrote: >>>> The very same, with no restriction of finiteness. Any T-riffic number >>>> has successor. :) >>> Well, 0 is finite, and the successor of 0 is finite, and the successor >>> of any finite in N is just another finite in N. Therefore N must >>> contain only finite naturals. >> No, the successor to an infinite, after an infinite number of >> successions from 0, is infinite. > > But that does not occur in N, but after N. N is defined as minimal, so > it cannot contain those infinites. >>> It's sporting of you to drop the requirement that all the naturals in N >>> have to be finite, but since all of them are, it's meaningless to say >>> "with no restriction of finiteness". That's kind of like saying N >>> contains all naturals "with no restriction of non-integer values". >>> I can say that, but it does not change the fact that all the members >>> of N are integers. >> Is the successor to ...11110000 not equal to ...11110001? > > That is no more relevant than asking whether the successor of -9 is -8. > >>> So I ask again, where are those infinite naturals and reals you keep >>> talking about? It's obvious they are not in N. >>> >> Not it's not. > > It is to anyone who knows how N is defined. > There are inductive sets. Every limit ordinal is one. > > But there are limit ordinals which are not minimal among limit ordinals. > > There is a unique minimal limit ordinal, and that is the set we call N. > > Every member of that set is a finite ordinal, and every non-empty finite > ordinal has an immediate predecessor. > > Since ordinals are, by definition, well ordered, they cannot contain and > endlessly decreasing sequences, which TO's models require. Neither can the reals. |