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From: William Hughes on 26 Oct 2005 00:35 Tony Orlow wrote: > David R Tribble said: > > Tony Orlow wrote: > > >> So, which element of the power set does not have a natural mapped to it? > > > > > > > Virgil said: > > >> {x in S:x not in f(x)} > > > > > > > Tony Orlow wrote: > > > Oh yeah, the entire set, last element and all. What element was that again? > > > > Your mapping does not include any natural that maps to the entire set > > *N, which it must do in order to be called a bijection, since *N is one > > of the members of P(*N). Show us that one single mapping, please. > It would obviously be the natural with an infinite unending strings of 1's, > right? Ah, but you want to know, how MANY 1's? No, this is irrelevent. No natural has a binary representation that consists only of 1's. > > > > But none of us can figure out where you keep pulling this "last > > element" gibberish from. Just let it go, Tony. > The entire set includes the last element, Tut tut. You have not been doing your exercises. Please repeat 150 times after lunch: Some sets do not have a last element. - William Hughes
From: albstorz on 26 Oct 2005 07:05 David R Tribble wrote: > David R Tribble: > >> Obviously, you do not understand. > >> My set is: > >> S = {0, 2^0, 2^2^0, 2^2^2^0, 2^2^2^2^0, ...} > >> S = {0, 1, 2, 4, 16, 65536, ...} > > Albrecht Stortz: > > I don't know what kind of math you apply here. Tribble-O-Math? > > I don't want discuss your very interesting system. > > Look at my starting posting if you want to know in what I'm interested > > to discuss. > > Start a new thread if you search for people which want talk about > > Tribble-O-Math. > > It's obvious that you don't understand simple arithmetic. 2^x is 2 > raised to the x power. 2^2^x is 2^(2^x), using ordinary arithmetic. > > I'm giving you set S so that you can tell us whether it is an > infinite set or not. You said: > > >> either there are infinite natural numbers or there is no infinite set. > > So I'm giving you set S, which obviously does not contain any > infinite numbers. So by your rule, the set is finite, right? There is no fundamental difference between your set S and any other infinite set, e.g. N. My method is to analyse the basic structures instead complexe ones which may hide the basic aspects which are in consideration. Talk about N, that is good enough, I think. Regards AS
From: albstorz on 26 Oct 2005 09:58 Tony Orlow wrote: > albstorz(a)gmx.de said: > > > > David R Tribble wrote: > > > David R Tribble wrote: > > > >> I've got a set S = {0, 2^0, 2^2^0, 2^2^2^0, ...}, which contains > > > >> all the powers of 2 of the form 2^p, where p=0 or 2^q. > > > >> 1) If it is not an infinite set, tell me how many members it has. > > > >> 2) If it is an infinite set, tell me what the smallest (first) > > > >> infinite number is a member of it. > > > > > > > > > > Albrecht Storz wrote: > > > > A short view upon this makes me think that you are writing sensless > > > > symbols. S don't contain all numbers of the form 2^2^q if you think > > > > about the sequence 0, 2^0, 2^2^0, 2^2^2^0,... . If this is your > > > > intention you may have a infinite sequence 0, 1, 1, 1, ..., and a set > > > > {0,1}, so > > > > 1) 2 > > > > 2) ? > > > > > > Obviously, you do not understand. > > > > > > My set is: > > > S = {0, 2^0, 2^2^0, 2^2^2^0, 2^2^2^2^0, ...} > > > S = {0, 1, 2, 4, 16, 65536, ...} > > > > > > > > > I don't know what kind of math you apply here. Tribble-O-Math? > > I don't want discuss your very interesting system. > > Look at my starting posting if you want to know in what I'm interested > > to discuss. > > Start a new thread if you search for people which want talk about > > Tribble-O-Math. > > > > Regards > > AS > > > > > This actually is interesting stuff, called tetrations. It is closely related to > the baseless enumeration of the reals, which I am finishing writing up (should > be posted soon). As I understand it, these kinds of formulas are not well > understood yet. Kind of hard to figure an inverse to this function. > -- > Smiles, > > Tony Thank you for this information. I had not know the name "tetration" before. I am very surprised to see that people work on this because I had studied this operation 30 years ago. But since I was a child and I had found that the operation don't lead to a new kind of numbers, I had lost my interest. I will read the information on wikipedia about this subject now. Regards AS
From: Tony Orlow on 26 Oct 2005 10:03 David Kastrup said: > Tony Orlow <aeo6(a)cornell.edu> writes: > > > David Kastrup said: > > > >> Every member "counts" itself. And every member "counts" a set > >> which ends with itself. The set of natural numbers does not end > >> with any member, and so it is not "counted" by any of its members. > >> Only the last member of such a set "counts" the set, and there is > >> no last natural number to "count" it. So you have the options of > >> declaring that the set can't be counted, or you have to invent an > >> unnatural number which, while not counting the set in the customary > >> way, is supposed to represent the count of the members of a set > >> obeying the Peano axioms. Not by actually doing the counting, but > >> by checking whether the axioms hold, and then declaring the count > >> of the set to be aleph_0, by decree. > >> > >> This does not make aleph_0 a member of the set of naturals. > > > > A decree does not make it correct either. > > Well, one knows that there is no natural number depicting the size. > > > The first option is better, to admit that the set size is equal to > > the largest member, but that there is no identifiable largest member > > and therefore no idnentifiable set size, which makes sense given the > > lack of endpoint for measure. > > This has, of course, the advantage that one need not deal with > number-like entities which behave rather peculiarly when doing > arithmetic with them. > > It has the disadvantage that there happens to be a hierarchy of > surjectability even between infinite sets. And those sets can be > compared in the context of surjectability, and it turns out that one > can group them in equavalence classes. > > And that makes it convenient to also have a _name_ for the cardinality > of the naturals and other infinite sets. This name can be used for > sorting; and a few rules concerning the arithmetic can be derived, > too, when deriving those rules from what happens to cardinality when > you form the union of disjoint sets. > > It _is_ a valid stance to leave the cardinality unnamed or unspecified > or unidentifiable or whatever else. But it is stopping short of what > _can_ be achieved in a consistent manner. > > Perhaps. Cardinality is better than nothing. And yet, I think that noting that the size really is not a number with an absolute value but needs to have some value applied to it in order to measure it, leads to more fruitful ways of dealing with the set. If the set cannot really have a size, then pretending it does seems like a waste of time to me. I suppose it leads to some broad classifications, but nothing like the precision of calculus or infinite series (convergent, anyway). So, I am trying to apply some more precise measures. I hope that's okay, even if it sounds like babbling hogwash. Does deep hogwash run still? Hmmm.... -- Smiles, Tony
From: Randy Poe on 26 Oct 2005 10:03
albstorz(a)gmx.de wrote: > David R Tribble wrote: > > David R Tribble: > > >> Obviously, you do not understand. > > >> My set is: > > >> S = {0, 2^0, 2^2^0, 2^2^2^0, 2^2^2^2^0, ...} > > >> S = {0, 1, 2, 4, 16, 65536, ...} > > > > Albrecht Stortz: > > > I don't know what kind of math you apply here. Tribble-O-Math? > > > I don't want discuss your very interesting system. > > > Look at my starting posting if you want to know in what I'm interested > > > to discuss. > > > Start a new thread if you search for people which want talk about > > > Tribble-O-Math. > > > > It's obvious that you don't understand simple arithmetic. 2^x is 2 > > raised to the x power. 2^2^x is 2^(2^x), using ordinary arithmetic. > > > > I'm giving you set S so that you can tell us whether it is an > > infinite set or not. You said: > > > > >> either there are infinite natural numbers or there is no infinite set. > > > > So I'm giving you set S, which obviously does not contain any > > infinite numbers. So by your rule, the set is finite, right? > > > There is no fundamental difference between your set S and any other > infinite set, e.g. N. Um, yes there is, e.g. R. Unless you can define the successor operation for x in the reals so that R can similarly be defined as some generator (e.g. 0) and its predecessors and successors. > My method is to analyse the basic structures Your method is to complain an call things you don't understand "contradictory", when the only thing they contradict is your own faith-based assertions. - Randy |