Cantor Confusion
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From: David Marcus on 6 Nov 2006 00:50 MoeBlee wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: > >"Foundations of Set Theory", North Holland, Amsterdam (1984). > > Here's a post by mueckenh(a)rz.fh-augsburg.de from a thread that is now > not open to reply: > > [begin post] > > David R Tribble schrieb: > > > Yes, I can see now that these are all finite sets. > > > And which are proper subsets of infinite sets. The set of all naturals > > that have been written now, for example. Obviously it's an ever > > growing set as time goes on, and will never contain the entire set > > of naturals that are possible. So it's simply a finite subset of N, > > and always will be. > > > Somehow you are using this fact to "prove" that N can't exist, perhaps > > employing some marvelous mathematical logic that has not been > > tainted by mainstream teachings. You show several finite sets. > > How do they prove anything about infinite sets? > > "a reasonable way to make this conform to a platonistic point of view The phrase "platonistic point of view" makes it pretty clear that they are discussing philosophy. > is to look at the universe of all sets not as a fixed entity but as an > entity capable of 'growing', i.e. we are able to 'produce' bigger and > bigger sets" [A.A. FRAENKEL, Y. BAR-HILLEL, A. Levy: Foundations of Set > > Theory, 2nd ed., North Holland, Amsterdam (1984) p. 118]. > > Why should simple infinite sets exist in another way? Just because > there is an axiom which cannot be satisfied like the axiom that there > be a straight bent line? > > Regards, WM > > [end post] > > If I undestand WM correctly, he's arguing that infinite sets can only > exist as described - capable of "growing". If that is not his argument > and if that is not the purpose of his adducing this quote by Fraenkel, > Bar-Hillel, and Levy, then I welcome him correcting any > misunderstanding I have about what WM is trying to say. > > I admit that I do not fully master the philosophical remarks by > Fraenkel, Bar-Hillel, and Levy, but I think I get the general idea, and > it is not suggestive that the sizes of sets themselves are variable. > The context of the quote is a discussion of the pros and cons of an > axiom of restriction, which would, roughly speaking, stipulate that > there are no sets except those whose existence follows from the axioms. > And the context includes discussions of various proposed axioms to > "restrict" the universe of sets. So different axioms give us various > possible larger or smaller universes, or some universes that include > some sets and other universes that exclude certain of those sets but > include certain others. So that is the sense of "growing" or "producing > larger and larger", which is to say that we can decide which axioms we > are to adopt and such decisions will determine whether or not the > universe includes or excludes certain sets. That does not suggest that > Fraenkel, Bar-Hillel, and Levy consider that a set itself can grow or > have different members at different times or anything like that. For > such axiomatic set theories, membership in a set is definite and sets > are determined strictly by membership. And Fraenkel, Bar-Hillel, and > Levy do not dispute that. > > MoeBlee -- David Marcus
From: mueckenh on 6 Nov 2006 08:47 Dik T. Winter schrieb: > In article <1162557178.206693.187650(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > I think we have obtained agreement, that we disagree in this point and > > that further discussion will not lead to a result acceptable for both > > parties. The final state is: > > > > 1) The number of columns is equal to the number of lines. > > 2) Both are infinite, according to set theory this means omega or > > aleph_0. > > > > But you believe: omega or aleph_0 is the maximum of the set of lines > > and the supremum of the set of columns, which, however, is not taken by > > any column. You believe the different meaning is not important. > > Again you misinterprete what I am saying. It is clear that there will > never be agreement if you continue misinterpreting what is written. > Omega (or aleph_0) is *not* the maxium of the set of lines. It is the > supremum of the cardinalities of the initial finite segments of the > set of lines. There is *no* position omega taken by any line. > > > I believe that a taken maximum and a not taken supremum are so > > different (although the difference is very tiny) that set theory, > > necessarily assuming this, is wrong. > > There is not a taken maximum. Oh no? The lengths of the columns of the list 0.1 0.11 0.111 .... are not omega? I was under the impression that there are aleph_0 lines with less than aleph_0 columns. > > > The difference between maximum und supremum is, by the way, the reason > > why I insist that an actually infinite set of natural numbers must > > contain a non-natural number. > > Yes, you keep insisting that, but as there is no maximum, that insistance > is futile. The number 0.111... or the diagonal has alep_0 1's as has the n-th column but there i sno line having aleph_0 1's. > But let's take the actually infinite set of natural numbers. > Suppose it contains your non-natural number. Now remove that non-natural > number from that set. Isn't the set still actually infinite? That shows only the self-contradiction of the notion actual infinity. > > What you are confused about is that the cardinality (and ordinality) of > the set of natural number is indeed actually infinite; but the *contents* > are only potentially infinite. You see not difference between the infinite man 1's of 0.111... and the finite many 1's in every line? > > > > > 0.1 > > > > 0.11 > > > > 0.111 > > > > ... > > > > > > > > you remember? Without an infinite number in the list there is no > > > > infinite diagonal defined. > > > > > > I state (you know that) that the diagonal: 0.111... > > > > Yes, again mixing up maximum and supremum. > > No, not mixing at all. Supremum only. Then 0.111... is not different from the finite sequences of 1's? > > > > > Fine. But the width is finite by definition. (We do not put finite > > > > segments together, but we have only finite seqments.) > > > > > > Nope. The width is unlimited, as is the length. And so is not finite. > > > > But the length is realized as a whole number by a set of elements > > (lines) while the width is not realized by a set of columns. > > You are confused between two concepts. The length is not realised in the > sense you think it is. The length every column is infinite and larger than any n while no line is. > > > > > > That is not contradicted. Width and length are equal. > > > > > > > > This amounts to say that there are infinite natural numbers or that the > > > > diagonal is longer than any line. > > > > > > Wrong. > > > > If you map the diagonal on a line by d_kk -> d_k, then the line (d_k) > > is longer than any line (a_mk) for any m e N. Therefore the diagonal > > has more elements than any line (if omega > m for all m e N). > > Right. The diagonal has more elements than any line. But not more than any column. Why is there a difference? > That is obviously > true by its definition. The sentence "I am greater than everybody including myself" cannot be true by definition. > Suppose it had equally many elements as some > line, than it would be shorter than the next line, and that next line > would not contribute to the diagonal. But your mapping above (d_kk -> d_k) > is not a mapping on a line, because there is no such line. > > > > > The maximum of a set of finite numbers which has no maximum is simply > > > > not present. > > > > > > Right. > > > > > > > It is *not* an infinite number which is larger than any > > > > finite number, because a maximum must belong to the set. > > > > > > Right. > > > > > > > And a supremum > > > > not belonging to the elements of the set does not yield a diagonal > > > > digit. > > > > > > And again right. And still your conclusion is wrong. Obviously the > > > diagonal has no finite length, because if it had it would have the > > > same length as one of the numbers, and then the next number would be > > > longer, which is a contradiction. So the diagonal is infinite in length. > > > > And nevertheless its length is assumed to be described by a > > (non-natural) number. (One cannot compare numbers with non-numbers.) > > Yes? What is the problem with that? The length of each and every line > is described by a natural number, and each and every natural number > describes a line. So if we want to describe the length of the diagonal, > it can not be a natural number. That length is called omega (if we want > to also maintain the ordering information) or aleph-0. But you don't see a difference between a diagonal having aleph_0 1's and the lines which it is constructed from which have less 1's? > > > > > Let us stick to Euclidean geometry. But that is unimportant. I see it > > > > is impossible to convince you of the existence of reality. > > > > > > What is the "existence of reality" in this context? > > > > That what exists, that was really is, contrary to a straight line which > > crosses itself. > > What do you mean with "that what exists, that what really is"? If you can't understand it then nobody can teach you. Regards, WM
From: mueckenh on 6 Nov 2006 08:56 Dik T. Winter schrieb: > In article <1162557998.763485.221280(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > > > No, something quite different was argued there. Namely that the limit > > > > > *also* is the number of edges in the infinite tree (or somesuch) > > > > > requires transfinite induction. > > > > > > > > But is does not. > > > > > > Indeed, it does not. But I thought you were maintaining that it would be? > > > > Good heavens! I should require transfinite induction? > > Sorry, I misread. You insist that the limit *also* is the number of edges > in the infinite tree. To prove that you need transfinite induction. Then you need transfinite induction to prove that the number of rationals in the case of an infinite set Q is countable. > > > > You need transfinite induction to show that "the number of edges in the > > > infinite tree" is equal to the limit of "the number of edges in the > > > finite tree". > > > > No. The number of edges in a finite tree is without interest. We > > consider only the infinite tree. The set of edges there can be > > enumerated like the set of rational numbers, for instance. If a mapping > > N --> Q is defined for every element of Q, then Q is countable. The > > mapping N --> {edges} has been established such hat every edge knows > > its number. > > You are wrong. Please try to understand the concept of countability. Only in the case of infinite sets it is of interest. Cantor defined it without (and before he was knowing anything about) transfinite induction. Regards, WM
From: William Hughes on 6 Nov 2006 09:36 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1162557178.206693.187650(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > I think we have obtained agreement, that we disagree in this point and > > > that further discussion will not lead to a result acceptable for both > > > parties. The final state is: > > > > > > 1) The number of columns is equal to the number of lines. > > > 2) Both are infinite, according to set theory this means omega or > > > aleph_0. > > > > > > But you believe: omega or aleph_0 is the maximum of the set of lines > > > and the supremum of the set of columns, which, however, is not taken by > > > any column. You believe the different meaning is not important. > > > > Again you misinterprete what I am saying. It is clear that there will > > never be agreement if you continue misinterpreting what is written. > > Omega (or aleph_0) is *not* the maxium of the set of lines. It is the > > supremum of the cardinalities of the initial finite segments of the > > set of lines. There is *no* position omega taken by any line. > > > > > I believe that a taken maximum and a not taken supremum are so > > > different (although the difference is very tiny) that set theory, > > > necessarily assuming this, is wrong. > > > > There is not a taken maximum. > > Oh no? The lengths of the columns of the list > > 0.1 > 0.11 > 0.111 > ... > > are not omega? I was under the impression that there are aleph_0 lines > with less than aleph_0 columns. Be careful with your quantifiers. The fact that every line has less than aleph_0 columns does not mean there are less than aleph_0 columns, it just means that no line has all columns. There are aleph_0 columns. Each line has less than aleph_0 columns. No line has all columns Every column is in some line. > > > > > The difference between maximum und supremum is, by the way, the reason > > > why I insist that an actually infinite set of natural numbers must > > > contain a non-natural number. > > > > Yes, you keep insisting that, but as there is no maximum, that insistance > > is futile. > > The number 0.111... or the diagonal has alep_0 1's as has the n-th > column but there i sno line having aleph_0 1's. > Correct, and completely irrelevent. > > > But let's take the actually infinite set of natural numbers. > > Suppose it contains your non-natural number. Now remove that non-natural > > number from that set. Isn't the set still actually infinite? > > That shows only the self-contradiction of the notion actual infinity. > > > > What you are confused about is that the cardinality (and ordinality) of > > the set of natural number is indeed actually infinite; but the *contents* > > are only potentially infinite. > > You see not difference between the infinite man 1's of 0.111... and the > finite many 1's in every line? I see a difference. However, I do not see a contradiction. 111... is not one of the lines and has a property that none of the lines share. > > > > > > > 0.1 > > > > > 0.11 > > > > > 0.111 > > > > > ... > > > > > > > > > > you remember? Without an infinite number in the list there is no > > > > > infinite diagonal defined. > > > > > > > > I state (you know that) that the diagonal: 0.111... > > > > > > Yes, again mixing up maximum and supremum. > > > > No, not mixing at all. Supremum only. > > Then 0.111... is not different from the finite sequences of 1's? Yes is it different. So what? The supremem of A may not be a member of A. The supremum of A can be different from every element of A. > > > > > > > Fine. But the width is finite by definition. (We do not put finite > > > > > segments together, but we have only finite seqments.) > > > > > > > > Nope. The width is unlimited, as is the length. And so is not finite. > > > > > > But the length is realized as a whole number by a set of elements > > > (lines) while the width is not realized by a set of columns. > > > > You are confused between two concepts. The length is not realised in the > > sense you think it is. > > The length every column is infinite and larger than any n while no line > is. Again this is true but there is no contradiction. The number of columns is not the maximum of the column lengths but the supremum. > > > > > > > > That is not contradicted. Width and length are equal. > > > > > > > > > > This amounts to say that there are infinite natural numbers or that the > > > > > diagonal is longer than any line. > > > > > > > > Wrong. > > > > > > If you map the diagonal on a line by d_kk -> d_k, then the line (d_k) > > > is longer than any line (a_mk) for any m e N. Therefore the diagonal > > > has more elements than any line (if omega > m for all m e N). > > > > Right. The diagonal has more elements than any line. > > But not more than any column. Why is there a difference? Because, by construction, every column is infinite and no line is. Why do you insist this simple fact is a contradiction? > > > > That is obviously > > true by its definition. > > The sentence "I am greater than everybody including myself" cannot be > true by definition. > > > Suppose it had equally many elements as some > > line, than it would be shorter than the next line, and that next line > > would not contribute to the diagonal. But your mapping above (d_kk -> d_k) > > is not a mapping on a line, because there is no such line. > > > > > > > The maximum of a set of finite numbers which has no maximum is simply > > > > > not present. > > > > > > > > Right. > > > > > > > > > It is *not* an infinite number which is larger than any > > > > > finite number, because a maximum must belong to the set. > > > > > > > > Right. > > > > > > > > > And a supremum > > > > > not belonging to the elements of the set does not yield a diagonal > > > > > digit. > > > > > > > > And again right. And still your conclusion is wrong. Obviously the > > > > diagonal has no finite length, because if it had it would have the > > > > same length as one of the numbers, and then the next number would be > > > > longer, which is a contradiction. So the diagonal is infinite in length. >
From: mueckenh on 6 Nov 2006 10:06
Dik T. Winter schrieb: > In article <1162559082.298879.296060(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > Because it is unrelated with the question where 1/oo was defined. > > > But to answer, I do not know who and when the definition of ordinal > > > number was introduced. I know *why* it might have been introduced: > > > to remove possible confusion. I have no idea what you mean with > > > "contrary meaning". As your statement A n: n < omega does not > > > indicate where n is coming from, I can not state whether it is > > > correct or not. But if the natural numbers are meant, than it is > > > certainly correct (if we assume the natural comparison between ordinal > > > numbers and natural numbers). > > > > n e N was implied. If you can compare two entities in some dimension > > like length or weight , then they are of the same sort with respect to > > this dimension. > > Perhaps. But we do not talk here about length or weight. We are talking > mathematics. How else lengths and weight could be compared if not by mathematics? > You can compare two entities of a different kind if, and > only if, a comparison function is defined. But as the natural numbers > can be embedded isomorphically in the ordinal numbers, there is a > natural comparison function. Why then do you oppose to state omega > n e N and to draw conclusions from that. > > > > > If omega is a number > n, then 1/omega is a number < 1/n. For all n e > > > > N. > > > > > > First, how do you *define* 1/omega? You know how arithmetic normally is > > > defined? But of course, once you define it, some properties of arithmetic > > > are lost. What is omega * (1/omega)? > > > > This property is lost. > > Still, how do you *define* 1/omega? You just state 1/omega = 0? Oh, well. I state omega > 2 which implies1/omega < 2. Then I take 3 instead of 2. And so on. > Perhaps. That makes you indeed lose a few arithmetical properties, so > you can not use them in general, but must use them with extreme care. The property above is not lost. > > > > > This definition has been conserved up to our days: Limesordinalzahl or > > > > limit ordinal number. > > > > > > Yes. But "limit" has not really been defined. > > > > Grenze ist immer an sich etwas festes, unveränderliches, daher kann > > von den beiden Unendlichkeitsbegriffen nur das Transfinitum als seiend > > und unter Umständen und in gewissem Sinne auch als feste Grenze > > gedacht werden. > > Still, I see *no* definition of "limit". Small wonder. > > > > > In ZF. > > > > > > > > Karel Hrbacek and Thomas Jech: "Introduction to set theory" Marcel > > > > Dekker Inc., New York, 1984, 2nd edition, p. 2: "So the only objects > > > > with which we are concerned from now on are sets." > > > > > > Again, depends on the model. > > > > Of course. But the usual set theory without yet specifying any models > > starts from this point. > > Not when I followed the courses on set theory. Why didn't you? Regards, WM |