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From: Virgil on 27 Oct 2005 21:05 > Albrecht Storz 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. Albrecht is arguing that if every A is a B, then it must be the case that every B is also an A. That every natural represents a set does not imply that every set represents a natural. Albrecht would fail any logic class with such a foolish claim.
From: albstorz on 28 Oct 2005 04:42 David R Tribble wrote: > > Consider the set of reals in the interval [0,1], that is, the set > S = {x in R : 0 <= x <= 1}. The elements of this set cannot be > enumerated by the naturals (which is why it is called an "uncountably > infinite" set). But all sets have a size, so this set must have a > size that is not a natural number. It is meaningless (and just > plain false) to say this set "has no size" or "is not a set". I'm not shure if the reals build a set in spite of you and Cantor and others are shure. A set is defined by consisting of discrete, distinguishable, individual elements. Now tell me: what separates a point on a line from the very next point on the line to be discrete? What separates sqrt(2) from the very next real number to be discrete? If you look only on individual points, you may have a set. But if you look on all of them? So, your above argumentation has no relevance to me. Proof the reals to be a set, then let's talk again. Regards AS
From: David Kastrup on 28 Oct 2005 05:04 albstorz(a)gmx.de writes: > David R Tribble wrote: > >> Consider the set of reals in the interval [0,1], that is, the set S >> = {x in R : 0 <= x <= 1}. The elements of this set cannot be >> enumerated by the naturals (which is why it is called an >> "uncountably infinite" set). But all sets have a size, so this set >> must have a size that is not a natural number. It is meaningless >> (and just plain false) to say this set "has no size" or "is not a >> set". > > I'm not shure if the reals build a set in spite of you and Cantor > and others are shure. A set is defined by consisting of discrete, > distinguishable, individual elements. Now tell me: what separates a > point on a line from the very next point on the line to be discrete? You really like to fantasize about things like the last natural, or the next real point, don't you? There is no next real point in a continuum. > What separates sqrt(2) from the very next real number to be > discrete? There is no very next real number. > If you look only on individual points, you may have a set. But if > you look on all of them? The decisive feature of a set is that for any of its members, you can verify its membership. A set is not required to be orderable, listable, arrangeable, or anything like that. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Robert J. Kolker on 28 Oct 2005 06:08 albstorz(a)gmx.de wrote: > > > I'm not shure if the reals build a set in spite of you and Cantor and > others are shure. > A set is defined by consisting of discrete, distinguishable, individual > elements. Now tell me: what separates a point on a line from the very > next point on the line to be discrete? There is no very next point under the ordinary ordering of reals. But given a pair of reals they are either equal or not. Between any two distinct real numbers there is always a third real different from the two given (with respect to the standard ordering of the reals). Bob Kolker
From: albstorz on 28 Oct 2005 07:20
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. > >> 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. Only one hundred years. What's that in relation to infinity? > > 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.) I don't know N+. I suppose you consider N instead of N_0. > > 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. Define "size". If size means anything to sets it should mean the number of elements in the set. If there are elements in the set, there should be a natural number which describe the size of the set. Now, if there are elements in the set, but they are uncountable many, the word "size" could only be understand in a different way than in the finite case. So you are right. (Assuming that a collection of infinite things is sensful to depict as a set since you can't construct an infinite set, you only can declare it. There is no connection betweeen every "regular set" and your infinite sets.) > > > > 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.) There is no infinite natural number since infinity is unconstructable, uncountable, unexplorable, unnameable, unapproachable, etc. > > > > 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. Yes > > > > 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 assume, there are infinite sets, in the sense that they are actually complete, there must be a natural number which is infinite, since natural numbers are sets. An actual complete set has an infinite number of elements. But then you have a problem, since natural numbers can't be infinite by definition. So you have to expand the definition of natural numbers or you have to deny the existence of infinte sets. That's your problem, not mine. > > 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. Either way, set theory with infinite sets is inconsistent. > > > > 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. So there is no actual complete infinte set. > > > > 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. > > What's the reason of an actual complete set 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. > There is no assumption by me. > > > 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. I've done no assumption. I use the assumptions of set theory and peano axioms. > > > > 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 more than endlessly, eternal, uncountable, infinite, .... > > There is nothing illogical or inconsistent about saying that some > sets contain more members than any finite number. More than finite is infinite. But infinite is not a size. It's a state. It's a word to depict the beginning of the unknown, the border of our understanding. > > > > 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. Set theory is inconsistent. > > > > 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). There is no amount "infinity". > > > > 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. For the finite case: yes. > > > > 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). Oh, it's just an accident of speaking to say: infinite many things are uncountable many things? You should have trust in the wisdom of language. Infinity is not greater than anything in any sense. It's just another way to say: unnameable, nameless great. Regards AS |