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From: David Kastrup on 27 Oct 2005 11:11 albstorz(a)gmx.de writes: > Randy Poe wrote: >> 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. > > Yes, it must. You can derive from the peano axioms or settle an > equal definition: nat. numbers are this, what count elements of > sets. Nonsense. The Peano axioms don't define what a "set" is. > If there is a congregation which elements you can't count you have > either no size or no set. You have no size that would be a natural number. And indeed you haven't. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Tony Orlow on 27 Oct 2005 11:15 imaginatorium(a)despammed.com said: > Tony Orlow wrote: > > William Hughes said: > > > > > > 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. > > I assume you mean in the infinite string, otherwise 1, 11, 111, etc would fit > > the bill. Why do you say I cannot declare that the 1's go on indefinitely? > > Oh, because mathematics does not work by "declaring that P", where P is > some proposition you'd like to be true. It's not a proposition. It's a bit string with a 1 for every natural number. This was requested, now it's delivered and refused. This is the last time I deliver pizza to THIS address. > > I wonder if you've noticed that (with the possible exception of other > cranks) you are the only one who goes about "declaring" this and that, > or "applying" values to things you want to "measure". LOL that's a good one. You declare aleph_0 to be some smallest infinity, applying this "value" to a set with no size so as to measure it. Please! Don't make me wet my pants laughing. I'm the only one....heh! That was good. > > Brian Chandler > http://imaginatorium.org > > -- Smiles, Tony
From: Tony Orlow on 27 Oct 2005 11:21 Robert J. Kolker said: > Tony Orlow wrote: > > >> > > > > 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, > > You continually confuse "how many" with "how much". Pah! Bob, I equate "how many" with "how much" when there is a 1-1 correspondence between element value and element count, as is the case with the naturals. I am not confused. You are not paying attention. > The former is > denominated by cardinal numbers. That later by some form of measure. > Google <measure theory>. The simplest sorts of measures are length, > area, volume etc. and generalizations thereof. > > Mathematicians do not confuse these two kinds of quantities, but you do, > therefore you are not a mathematician. Your inability to learn from your > betters indicates strongly that you will never learn to be one. You have no clue what I'm doing here, do you Bob? The whole point is to bring these various disparate measures that don't get along, and turn them into a cohesive team that can discriminate all types of sets accurately. But I don't expect you to get that. What do you care about this topic anyway? It's not a physical matter, so to you, it doesn't exist. You know, like that nonexistent "mind". > > Before you criticize well established mathematical theories you should > learn what they are first. You have not done this and you show no > indication of ever doing so. I know what they are, but they don't make sense. > > Bob Kolker > -- Smiles, Tony
From: David R Tribble on 27 Oct 2005 11:34 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. >> There is no way you have an infinite number of them >> without achieving infinite values within the set. > David R Tribble wrote: >> Yes, you and Albrecht keep saying that repeatedly. Please demonstrate >> why it must be so, because it's not. > Albrect Storz wrote: > 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. No, I don't, since this has been established mathematics for well over a hundred years. You are making claims that contradict established mathematics, so it is your task to support your claims with proof. But anyway, it is trivially easy to show that no natural can represent the size of the set of naturals: Let N+ = {1,2,3,...}, the set of finite naturals greater than zero. (We'll use N+ instead of N because it seems to appeal to your sense of logic.) Suppose m in N+ represents the number of elements in N+. Since m is a member of N+, it must be a finite natural. Since m represents the number of elements in N+, each of which enumerates itself, m must be greater than all the other members in N+. But every finite natural has a finite natural successor, so m+1 must exist, and m+1 must also be a finite natural, and thus a member of N+. But m+1 is greater than m, which contradicts our supposition that m is greater than all other members in N+. Contradiction. This forces us to conclude that we cannot find any m in set N+ that represents the size of N+. Which in turn forces us to conclude that the size of set N+ is not a natural number. QED. > 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. You seem to be saying that infinite naturals cannot exist. If you really are saying this, then apologies to you. (But no apologies to Tony, who does not believe this.) > 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. Every set does have a specific number of elements. That's the cardinality of each set. > 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. Now you seem to be saying that infinite natural numbers do exist. If you mean that infinite natural numbers exist, then you must prove this to be so. (Not infinite ordinals or infinite cardinals, since we know that these exist, but infinite naturals.) Or if you are saying that there are no such things as infinite sets, then you must prove this to be so. Prove that what everyone else thinks is an infinite set (such as N) does not really contain an infinite number of elements and is actually finite. Either way, I withdraw my apology. > 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, This holds for all finite sets of finite natural numbers. This does not hold for infinite sets which have no largest member. > if there are infinite nat. numbers > there must be a infiniteth nat. number representing this set. That implies that there is a greatest member in the set that is an infinite natural. But this infinite set does not have a greatest member. So there is no reason for an infinite natural to exist. > 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. Your assumption that the infinite set contains an infinite natural is what leads to your contradiction. Since this assumption is false, there is no actual contradiction. > So either the definition of nat. numbers must be changed or there is no > infinite set of natural numbers. Or your assumption must be changed. > 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. "Infinity" has several meanings, and you're confusing at least two of them. An "infinite set" has a size represented by an infinite (transfinite) cardinal number. The limit of a series or sum that "approaches infinity" is a limit value that is larger than any real number. A "countably infinite" set is a set whose members can be enumerated (bijected) with the natural numbers. An "uncountably infinite" set is a set that has more members than the set of natural numbers. There is nothing illogical or inconsistent about saying that some sets contain more members than any finite number. > 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. I don't think I'm misinterpreting what you're saying. I think what you're saying is wrong. What Tony is saying is certainly wrong. > In this state there is no real problem with all this. aleph_0 is just > onother symbol for infinity. One particular kind of infinity, yes. > The problems occure in that moment if someone declares, that aleph_0 is > a size, which is greater than any nat. number. But that is the definition of Aleph_0; it is the size of the set of natural numbers (or any other set containing that amount of members). > 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, The very fact that a bijection exists between set A and set B proves that the two sets are the same size. This is true for all finite and infinite sets. Bijection is the obvious way to show that sets are the same size. > since infinity is something sizeless, endless, countless. > That's all. Well, infinity is certainly a kind of "endlessness", yes. But it's not countless (if it's a countable infinity), and it's certainly not sizeless (since infinity is greater than any finite).
From: David Kastrup on 27 Oct 2005 11:34
albstorz(a)gmx.de writes: > David Kastrup wrote: > >> Albrecht wrote: >> > 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. > > Wrong. That's like saying if a zoo containes animals, there must be > animals in the zoo. Think again. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum |