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From: David Kastrup on 19 Oct 2005 07:56 albstorz(a)gmx.de writes: > Virgil wrote: >> In article <1129684276.251366.121150(a)g47g2000cwa.googlegroups.com>, >> albstorz(a)gmx.de wrote: >> >> > David R Tribble wrote: >> > >> > > >> > > Because I can prove it (and it's a very old proof). A powerset of >> > > a nonempty set contains more elements that the set. Can you prove >> > > otherwise? >> > >> > This argument is stupid. Is there any magic in the powerfunction? >> >> "Proofs" are not stupid until they can be refuted. The proof that for an >> arbitrary set S, Card(S) < Card(P(S)) has not been refuted by anyone. > > > Even if you think that the powersets of finite and infinite sets > have both a greater cardinality than their starting sets, you would > not really think it depends on the same cause in both cases. > > You must proof it independently for finite and for infinite sets. In > this sense the argument is stupid. Uh, no. The proof depends merely on the fact that some value has to be either a member of a set, or not. And if some value is in one set, bur not another, then those two sets are different. That's all. Finiteness or infiniteness does not even play into it. The proof just constructs a set which differs by the membership of at least one particular value with every target set in the assumedly complete mapping of set to powerset. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: albstorz on 19 Oct 2005 07:59 David R Tribble wrote: > Albrecht S. Storz wrote: > >> [...] > >> Since there is no biggest number and since there is no infinite number, > >> the size of the set of numbers in form of sets of #s is undefined as > >> the biggest natural number is undefined. > >> > >> But the sequence of the sets of # fullfill the peano axiomes. So this > >> set must be infinite. > >> > >> The cardinality of a set is not able to be infinite and "not defined" > >> at the same time. > >> This is the contradiction. > > > > David R Tribble wrote: > >> I don't see the contradiction. The size of the set is "not defined" > >> to be the same as any natural number, and the set size is obviously > >> infinite. This is no contradiction, since no natural number is > >> infinite. > >> > >> The thing that is "not defined" is the largest natural, which obviously > >> does not exist. But the set size is infinite, and is nicely defined > >> by an infinite cardinal. > >> > >> You seem to be mixing the two concepts of "natural" and "cardinal" > >> numbers to create a supposed contradiction, but that does not work. > > > > Albrecht S. Storz wrote: > > You are not able to understand that there is no difference between > > numerals and sets. > > I have no problem seeing the correspondence between natural numbers > and von Neumann sets. But neither of these are the same as > cardinalities, which are not numbers, but measures (sizes) of sets. Natural numbers are sets. Why be so delicate about this? It's not only a correspondence between them. It's identity. Only a fool is unable to see that fact if the numbers are shown in unitary (1-adic) system. And now the cardinality. The definition of cardinality bases on sets. The existence of infinite sets bases on definition. The meaning of the cardinality of an infinite set is unknown. Coincidently natural numbers and cardinalities are undistinguishable in finity. Cardinality is just a artificial concept with no sens and meaning. Define the existence of unicorns and be happy. Regards AS > > > > My sketches shows this exactly. > > Cantor proofs his wrong conclusion with the same mix of potential > > infinity and actual infinity. But there is no bijection between this > > two concepts. The antidiagonal is an unicorn. > > There is no stringend concept about infinity. And there is no aleph_1, > > aleph_2, ... or any other infinity. > > For that to be true, there must be a bijection between an infinite > set (any infinite set) and its powerset. Bitte, show us a bijection > between N and P(N).
From: albstorz on 19 Oct 2005 08:30 David Kastrup wrote: > Tony Orlow <aeo6(a)cornell.edu> writes: > > > stephen(a)nomail.com said: > >> Tony Orlow <aeo6(a)cornell.edu> wrote: > >> > stephen(a)nomail.com said: > >> >> albstorz(a)gmx.de wrote: > >> >> > >> >> > But there is a slight difference. Since there is no infinite > >> >> > natural in form of a set of Os and since after every set of #s > >> >> > there should be a O, the size of the set of the naturals as > >> >> > sets of #s could not extend the "biggest" number of the > >> >> > naturals in form of sets of Os. Since there is no biggest > >> >> > number and since there is no infinite number, the size of the > >> >> > set of numbers in form of sets of #s is undefined as the > >> >> > biggest natural number is undefined. > >> >> > >> >> Whoever said the size of a set has anything to do with the > >> >> "biggest" element? > >> > Stephen, did you even look at the diagrams he presented? Do you > >> > not see that the width of the square and the height are the > >> > same. Do you not see that the width is the count of naturals and > >> > the height is the value? The picture said so, that's who. > >> > >> What square? The sides of a square are line segments. The four > >> corners of the square are defined by the ends of those line > >> segments. If your lines extend indefinitely, then there is no > >> square. > >> > >> This is not a square: > >> +-----------..... > >> | > >> | > >> . > >> . > >> . > >> > >> A square has four corners. This only has one "corner". > >> Remember, infinite lines do not end. Not even "at infinity". > > (sigh) As Albrecht said, the square is defined by the diagonal at 45 > > degrees. > > There is no "diagonal" for something that has only one corner. > > > For every natural value represented by 0's in the diagram there is > > an equal count represeted by #'s. > > It does not make sense to talk about "an equal count" for things that > don't end. > > > This is the identity relationship between count and value that I've > > been talking about. Think of it as the limit of a square as the side > > goes to oo. Your objection is just another form of "No Largest > > Finite!! No Diagonal Corner!!! (jingle jangle)" Oh, nice wind > > chime!! > > Well, too bad that you insist on making the same mistake all over > again. Small wonder you get your nose rubbed into it all over again. > > -- > David Kastrup, Kriemhildstr. 15, 44793 Bochum You are not able to respond to my concept. You prefer to correct all over again the same mistakes (if you are shure to accord with the majority in this aspect). It's the usual dishonest of the dogmatic people. Or are you anxious to disgrace yourself? If you have two straight lines, suptending at point Zero, what is the rectangular distance from a point in infinity laying on the one straight line to the other? Regards AS
From: albstorz on 19 Oct 2005 08:38 William Hughes wrote: > albstorz(a)gmx.de wrote: > > <snip> > > > > > If we accept the uncountability as a form of infinity, this leads to > > the paradoxon that the natural numbers are not countable. > > No, the natural numbers are countable precisely because > they do count themselves. This is exact my argument: there are uncountable many natural numbers (nothing other means infinity many) but the natural numbers are shurely countable since they count themself. You are not able to recognise a paradoxon if you see it. > The fact that there is no > natural number that repsresents this "count" is not a paradox > because the "count" is defined in terms of bijections. [You > may not like the use of the terms "count" and "countable" > because you think they should imply something different. So > be it. However, you cannot say "you are using a term which > I think should mean somthing different, so you must mean > not what you mean but what I mean"] > > - William Hughes Regards AS
From: David Kastrup on 19 Oct 2005 08:42
albstorz(a)gmx.de writes: > You are not able to respond to my concept. You prefer to correct all > over again the same mistakes (if you are shure to accord with the > majority in this aspect). They don't go away by ignoring them. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum |