An uncountable countable set
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From: Virgil on 26 Jul 2006 23:01 In article <1153947394.828304.296770(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > > > But every natural is finite. Every digit which can be indexed by > > > naturals is *by definition* in the true list (not the extended list) > > > > The "true list" of all indices is the one represented by the unending > > "0.111...", since every ending list is too short. > > My true list is unending. Every n is finite. N is infinite. My list > contains only finite numbers. But my list is infinite. My list *is* N. > More natural numbers are not available. More indices are not > available. > > > > and every digit which cannot be indexed is not in the true list. So > > > your attempt to explain that a number which is not in the true list > > > nevertheless could be indexed is completely unfounded. > > > > Thus "muecken" acknowledges the infiniteness of the set of indices( > > naturals) and simultaneously declares it finite. > > You intermingle magnitude and number of numbers. On the contrary, It is we who object to "mueckenh"'s mangling/\. WE note that every natural is finite but that there are more than any finite number of them. It is "mueckenh" who mangles that.
From: Virgil on 26 Jul 2006 23:03 In article <1153948290.908110.306190(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1153863159.167151.152210(a)h48g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Michael Stemper schrieb: > > > > > > > >Is July 11, 2002 contemporary enough? Thomas Jech: Set Theory > > > > >Stanford.htm, Stanford Encyclopedia of Philosophy: Until then, no one > > > > >envisioned the possibility that infinities come in different sizes, and > > > > >moreover, mathematicians had no use for "actual infinity." > > > > > > > > Okay, so your own reference shows that philosophers don't think that > > > > mathematicians use the term "actual infinity". That sounds correct. > > > > > > T. Jech is one of the leading set theorist. > > > He says up to the advent of set theory there was no use for actual > > > infinity. > > > > Does he say that there is no use for actual infinity within current set > > theory? I doubt it. > > He says that there is only use for actual infinity in (current and > uncurrent) set theory > > Regards, WM So that all of "mueckenh"'s railings against actual infinities are thereby countered.
From: Virgil on 26 Jul 2006 23:15 In article <1153948472.375805.93600(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > > > > > > > The proof is given by the list: > > > > > > > > > > > > > > 1 0.1 > > > > > > > 2 0.11 > > > > > > > 3 0.111 > > > > > > > ... ... > > > > > > > w 0.111... > > > > > > > > > > > > That is neither a proof, nor a list. > > > > > > > > > > The list is given in the upper part. Call the whole an extended > > > > > list. > > > > > It shows that w is not the number of the naturals, because w > > > > > appears in > > > > > the sequence of ordinal numbers only once. > > > > > > > > That is only a statement, *not* a proof. > > > > > > Why is the ordinality of the left column of the extended list another > > > than the ordinality of the right column? > > > > I do not state so. The ordinality of both "lists" is w+1. Why do you > > think > > they are different? > > Look at the list. The ordinality of the number > 1 > 2 > 3 > ... > n > > is the same as that of the unary number in the corresponding line > 0.111...1 (with indices 1,2,3,...,n) as long as n is a natural number. > But the ordinality of > 1 > 2 > 3 > ... > w > differs from the ordinality of the unary of the corresponding line > 0.111... > This proves that there is no possibility to interpret w as the number > which counts the naturals. That would only follow if it were necessary that 0.111... represent a natural number, but it does not and cannot. If one has 0. 0.1 0.11 0.111 .... and then one finishes off the sequence by appending a LUB of 0.111... What one has is that each element in that extended list represents the set of all its predecessors, just as do the ordinals up through omega each represents the set of all its predecessors. > > In general, the ordinal number of any initial segment of ordinal numbers > > is the smallest ordinal number that is larger than all ordinal numbers in > > that segment. (This can be done because the ordinal numbers form a > > well-ordered set.) And note that 0 is also an ordinal number (of the empty > > set). So the ordinal number of {} is 0, of {0, 1, 2} is 3, etc. > > Finally, the ordinal number of {0, 1, 2, 3, 4, ..., w} is w+1. And so > > is the ordinal number of {1, 2, 3, 4, 5, ..., w}, because there is an > > order preserving bijection between the two. > > > > You apparently did not appreciate that 0 is an ordinal number. (And that > > is independent of Bourbaki.) > > -- > > That makes no difference. > > 0 0. > 1 0.1 > 2 0.11 > 3 0.111 > > ... > w 0.111... > > The interruption between finite and infinite remains. But in the well ordered set described, each element corresponds to the ordinal formed by the ordered set of all prior ordinals. That "mueckenh" is too dim to distinguish between successor ordinals and limit ordinals is his cross-eyed bear, not ours.
From: Dik T. Winter on 27 Jul 2006 19:25 In article <1153948199.656603.23130(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > In article <1153829288.635491.96140(a)m73g2000cwd.googlegroups.com> muecken= > h(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > I need not count to w. If a column contains one 1 then the *+sum is 1. > > > This is a definition like that of Cantor's list. > > > > Depends. I have no idea (and it is very dissimilar), whatever way I > > look at it, I find: > > (*+){k = 1 .. n} Ak = An > > and I can not substitute oo for n (so as to include all Ak), because there > > is no Aoo. > > Therefore you cannot need and cannot use it. It is simply not > available. Yes, so it is not defined for the *+ sum of all An... > > > (every digit of n can be indexed) ==> (n is in the list) > > > > Where is the proof? > > It is even an equivalence by *definition*. The list contains all > natural numbers. All indexes *are* natural numbers and vice versa. I do not see the equivalence. The equivalence is: (every digit of n can be indexed) ==> (every index is in the list) I still see no proof that n is in the list, because I do not see a proof that n is an index. > > There is an easy proof of: > > (n is in the list) ==> (every digit of n can be indexed) > > but not of the converse. > > You cannot index other digits than those which are indexes. All indexes > are all naturals. All naturals are by definition in he true list. Yes. This does *not* proof that the *+ sum of them all (once defined) is in the list, because there is no proof that it is an index. Rather, there is an easy proof that it is *not* an index. > > > > > The last digit of 0.111... has no ordinal number. > > > > > > > > Indeed, but the reason is that there is no last digit. w is still a > > > > potential infinity, you can not count to it (for one thing, it has > > > > no predecessor). > > > > > > Why does the left column of my extended list differ from the right one? > > > There are the same numbers given, only their representation is > > > different. > > > > You do not show here what you are referring to, but I think I understand > > what you mean. > > Look at the list. The ordinality of the number > 1 > 2 > 3 > ... > n > > is the same as that of the unary number in the corresponding line > 0.111...1 (with indices 1,2,3,...,n) as long as n is a natural number. > But the ordinality of > 1 > 2 > 3 > ... > w > differs from the ordinality of the unary of the corresponding line > 0.111... > This proves that there is no possibility to interpret w as the number > which counts the naturals. Reread what I wrote. Your list does not contain ordinal numbers, but natural numbers. When you add 'w', you add an ordinal number, not a natural number. So why would you expect the same would hold? Now do the same with ordinal numbers only: 0 1 2 ... n and see the ordinal number of that list is n+1. (And, yes, 0 is an ordinal number, it is the ordinal number of the empty set.) > > > The least ordinal of a class is also its cardinal number. If you say N > > > has the cardinal number omega you are not wrong today. Even Cantor did > > > so at his later times. > > > > Perhaps. I do not know whether that is really true for all sets. But wh= > at > > you stated is still wrong. Formally aleph-0 is defined as the equivalence > > class of sets that biject with the set of naturals. w is defined as the > > order type of N (and because N is well-ordered), it also becomes the > > ordinal number of N. You need proof to show that all ordinal numbers of > > a class (what is a class?) > > A class of numbers is a set of ordinals of sets with same cardinality. Possible, that is a proper definition. > Their cardinality in newer set theory (and already by the late Cantor) > is expressed by the least ordinal. The cardinality of sets w , 2w, w^2, > ... is expressed by aleph_0 or by omega. Yes, already by the late Cantor. > >are for sets that are in the same equivalence > > class with respect to bijection. But perhaps I am wrong, but I think that > > today the distinction between w and aleph-0 is made more clear than in > > Cantor's later times. > > On the contrary! Cantor distinguished very clearly in all his written > papers. Except in his later times, as you state so eloquently just above. Or do you *not* see a contradiction between these two statements? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 27 Jul 2006 19:36
In article <1153948472.375805.93600(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > Why is the ordinality of the left column of the extended list another > > > than the ordinality of the right column? > > > > I do not state so. The ordinality of both "lists" is w+1. Why do you > > think they are different? See also my previous comment on this. > Look at the list. The ordinality of the number > 1 > 2 > 3 > ... > n Ordinal number n. > is the same as that of the unary number in the corresponding line > 0.111...1 (with indices 1,2,3,...,n) as long as n is a natural number. It is the same. > But the ordinality of > 1 > 2 > 3 > ... > w Ordinality w+1. > differs from the ordinality of the unary of the corresponding line > 0.111... But the *list* 0.1, 0.11, 0.111, ..., 0.111... has ordinality w+1. You stated that the ordinality of the lists was different, but that is false. It is only when you take initial segments of the ordinal numbers that you *always* find that the ordinal number of the segment is one more than the last ordinal. > This proves that there is no possibility to interpret w as the number > which counts the naturals. And this interruption is independent of the > first ordinal, whether you may choose 0 or 411 as the first one. Try it starting with 0 (actually in truth also the first ordinal), and see how it works out. > If you > arrange the list such that the unary numbers are the same as the > numbers in the left column, then it is impossible to continue into the > infinite. Here is an example starting with 3: > > 3 0.111 > 4 0.1111 > 5 0.11111 > ... > w 0.111... I have no idea what this is supposed to prove. Both the left "list" and the right "list" have ordinality w+1. This is just because the lists {0, 1, 2, ...}, {1, 2, 3, ...} and {3, 4, 5, ...} all have the same ordinality, i.e. w. > That makes no difference. > > 0 0. > 1 0.1 > 2 0.11 > 3 0.111 > > ... > w 0.111... > > The interruption between finite and infinite remains. *What* interruption? In this case when you take an initial segment of your "list" the ordinal number of that list is the last number plus 1. As is the case when you take the complete list. So: {0, 1, 2, 3} has ordinal number 4, which can be represented as 0.1111 (which is not in that list) and your complete list has ordinality w+1 (for which I would not even venture to guess at a unary representation). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |