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From: Virgil on 26 Oct 2005 18:01 In article <1130352870.248287.100470(a)f14g2000cwb.googlegroups.com>, "David R Tribble" <david(a)tribble.com> wrote: > 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)} > > > > David R Tribble said: > >> 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? > > Perhaps 'N' ones? > > So your infinite natural that maps to the entire set *N is: > s = 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + ... > s = 1 + 2 + 4 + 8 + 16 + ... > > Thus s is a sum of an infinite number of finite powers of two (2^p), > each term being twice the previous term. So s is your 2^N-1, where > N is your "unit infinity": > N = 1 + 1 + 1 + 1 + 1 + ... > > (We'll pretend for the moment that s and N are different, and ignore > the fact that they are actually the same value.) > > So the subset that s maps to is: > Ps = { 0, 1, 2, 3, ... } > > Ps should look familiar - it's just N, the set of finite natural > numbers. So you can see that s, an infinite number, maps to the > infinite set of all finite naturals, or N. But s does not map to > *N, because Ps does not contain any infinite naturals as members. > > It's fairly easy to show this to be the case using TOmatics. > We choose an s of all 1 bits, N bits in length, as the infinite > natural to map to Ps, where each natural member k in Ps corresponds > to the 2^k bit of s. If s is composed of N bits, then Ps contains > N members. But s itself cannot be a member of Ps, because the > members of Ps are all less than log2(N) (which is also an infinite > value). > > Fine, you say, let each member in Ps have N bits. But then s must > now have 2^N bits, and s is still too big to be a member of Ps. > You can keep doing this and choose even longer bitstring lengths > for s and the members of Ps, but you'll find an s that is a member > of Ps, all because N < log2(N) for any value of infinite N. I think that inequality symbol is reversed from what it should be, at least if one expects to have 2^N > N. > > We are forced to conclude that there is no natural s that maps to > *N, and that therefore your mapping scheme is not a bijection > between *N and P(*N). And it never was, and never will be, either!
From: David R Tribble on 26 Oct 2005 18:56 David Kastrup said: >> This does not make aleph_0 a member of the set of naturals. > Tony Orlow wrote: > A decree does not make it correct either. 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. Why is this better than just saying that the size of the set is larger than any element in the set? Every set has a size, but not every set size is a natural number.
From: albstorz on 27 Oct 2005 10:30 David R Tribble wrote: > David R Tribble said: > >> So I'm giving you set S, which obviously does not contain any > >> infinite numbers. So by your rule, the set is finite, right? > > > > Tony Orlow wrote: > > If it doesn't contain any infinite members, it's not infinite. Those terms > > differ by more than a constant finite amount, but rather a rapidly growing > > amount greater than 1. There is no way you have an infinite number of them > > without achieving infinite values within the set. > > Yes, you and Albrecht keep saying that repeatedly. Please demonstrate > why it must be so, because it's not. Your argumentation is not fair, but I don't wonder about that. _You_ has to show, that in the case of the whole set there is no natural number as big as the whole set. You argue: there is no infinite natural number since the peano axioms don't allow an infinite natural number. That's right. I agree with you. But that's no proof about sets. That's only an aspect of the definition which contradicts with the fact, that every set has a number of elements. You misinterpret totally when you say, I think there must be an infinite natural number. I don't think so. I only argue that, if there are infinite sets, there must be infinite natural numbers (since nat. numbers are sets). I don't say: there are infinite sets. You say: there are infinite sets and there is no infinite number. And I say: If there are infinite sets there must be infinite numbers. My argumentation is very easy: Every nat. number represents a set. If you look at the first 100 nat. numbers, the 100th nat. number "100" represents the set {1, ... , 100}. As this holds for every nat. number, if there are infinite nat. numbers there must be a infiniteth nat. number representing this set. But the definition of the nat. numbers with complete induction leads to the consequence, that there could not be an infinite nat. number. That's the contradiction. So either the definition of nat. numbers must be changed or there is no infinite set of natural numbers. Or infinity must be interpreted in a completely other way. Not as a size like you do. Infinity is just an unability to count it with numbers because it runs out of all what we can know. All this is shown very expressive in my sketches at the start of this thread. Why do you misinterpret all the time? Maybe my ability to express my thoughts in english is too bad. But why do you misinterpret Tony also? I think he is native english speaker and you should be able to understand him. In this state there is no real problem with all this. aleph_0 is just onother symbol for infinity. The problems occure in that moment if someone declares, that aleph_0 is a size, which is greater than any nat. number. But there is no "greater" or "less than" or something like this. There is just something other, something out of the things we could measure, wigh or count. The possibility of bijection don't say anything about the size of infinity, since infinity is something sizeless, endless, countless. That's all. Regards AS
From: Randy Poe on 27 Oct 2005 10:39 albst...(a)gmx.de wrote: > My argumentation is very easy: > Every nat. number represents a set. If you look at the first 100 nat. > numbers, the 100th nat. number "100" represents the set {1, ... , 100}. > As this holds for every nat. number, if there are infinite nat. numbers > there must be a infiniteth nat. number representing this set. No, there musn't. There is no logical basis on which to draw this conclusion from your premise. > But the definition of the nat. numbers with complete induction leads to > the consequence, that there could not be an infinite nat. number. Correct. Your first "conclusion" is merely an assertion, not based on axioms. The only rule you're using is "there must", the same thing Tony Orlow does: shout loudly when you don't have a mathematical basis for "must". There "must" based on WHAT? WHY must there? - Randy
From: David Kastrup on 27 Oct 2005 10:43
albstorz(a)gmx.de writes: > Your argumentation is not fair, but I don't wonder about that. > _You_ has to show, that in the case of the whole set there is no > natural number as big as the whole set. > You argue: there is no infinite natural number since the peano axioms > don't allow an infinite natural number. > That's right. I agree with you. > But that's no proof about sets. That's only an aspect of the definition > which contradicts with the fact, that every set has a number of > elements. You have a strange view of "fact". This is certainly not a fact, though it is desirable. And in order to give substance to that desire, cardinalities were invented, which are not really proper numbers. > You misinterpret totally when you say, I think there must be an > infinite natural number. I don't think so. I only argue that, if there > are infinite sets, there must be infinite natural numbers (since nat. > numbers are sets). That is like saying if there are animals in a zoo, there must be an elephant (since an elephant is an animal). Too bad you'll find zoos without an elephant in them. > I don't say: there are infinite sets. You say: there are infinite > sets and there is no infinite number. There is a zoo without an elephant in them (why, my home town has one). > And I say: If there are infinite sets there must be infinite > numbers. You say: if there is a zoo, there must be an elephant. > My argumentation is very easy: > Every nat. number represents a set. Every elephant represents an animal. > If you look at the first 100 nat. numbers, the 100th nat. number > "100" represents the set {1, ... , 100}. If you look at the first 10 elephants in zoo, the 10th elephant accounts with its predecessors for 10 animals. > As this holds for every nat. number, As this holds for every elephant. > if there are infinite nat. numbers If there is any number of animals. > there must be a infiniteth nat. number representing this set. there must be an elephant representing the animals. > But the definition of the nat. numbers with complete induction leads > to the consequence, that there could not be an infinite nat. number. > > That's the contradiction. Where is the contradiction? All numbers can be represented by sets, but that does not mean that all sets can be represented by numbers. > Why do you misinterpret all the time? Maybe my ability to express my > thoughts in english is too bad. Don't worry about that. You can't think straight in German, either. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum |