An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: mueckenh on 8 Aug 2006 16:11 Dik T. Winter schrieb: > > > You are claiming that K is in the list of indices. That makes no sense > > > at all. > > > > I am claiming that every index is in the list of indexes. This list is > > infinite. There are infinitely many indexes. And, yes, 0.111... is not > > among them and therefore cannot be indexed. > > No proof, yet. K = 0.111... can also be written as "for every n in N, > K[n] = 1". Every number which can be written in this manner is contained in the list, because there is every n and every digit which can be indexed. EVERY INDEX IS THERE. Every index position is there. It is the set of all indexes and o all index positions. But 0.111... is not there. Because this number cannot be indexed. Regards, WM
From: Virgil on 8 Aug 2006 16:22 In article <1155067631.761632.250240(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > But the *+ sum is defined for all digit positions which can be indexed > > > by a natural number. > > > > You still fail to see. There is a definition for (*+){k = 1 .. n} Ak, > > I still do not see a definition for (*+){k = 1 .. oo} Ak. Can you, > > please, once give a proper definition? > > There is no natural number oo. Therefore {k = 1 .. oo} is nonsense. > There is, as you say, a definition for (*+){k = 1 .. n} Ak. More is not > possible. Sure it is! There is a set, N, of natural numbers, the set of finite ordinals as defined by von Neumann, and we can easily define "{k = 1 .. n} Ak" to mean "{k in N\{0}} Ak". > > > > > n is a natural number. An index is a natural number. "Index" and > > > "natural number" are synonymous. > > > > You are assuming that only digits of natural numbers can be indexed. > > But 0.111... is not a natural number. And you are stating (in fact): > > (every digit of 0.111... can be indexed) ==> (0.111... is in the list). > > so, why does this follow? > > Because: index = natural number = "number that can be indexed" Non sequitur. That every digit can be indexed does not require that the string itself can be, or is, indexed. > > > > > > 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. > > > > > > Then it cannot be indexed. Then it contains digit positions which are > > > not indexed by natural numbers. > > > > You state so, and remain stating so, without any proof. > > It is a definition. What definition is that? I know of no definition which denies natural number indexability to every digit in an endless string of digits. If one can index every rational number, and one can, indexing every digit in a string is trivial by comparison. Except to those with severe mental blocks, like "mueckenh".
From: Virgil on 8 Aug 2006 16:24 In article <1155067888.355942.225340(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > EVERY INDEX IS THERE. Every index position is there. It is the set of > all indexes and o all index positions. But 0.111... is not there. > Because this number cannot be indexed. > > Regards, WM The "Hilbert Hotel" method allows one always to insert one more and still have rooms for all.
From: mueckenh on 8 Aug 2006 16:28 Virgil schrieb: > In article <1154967573.942270.13030(a)p79g2000cwp.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Example: > > The third 1 of 0.111... can index the third 1 of 3 = 0.111 as well as > > the third one of 5 = 0.11111. > > > Since the third 1 and the fourth 1 are identical, absent their > predecessors, neither can index anything by itself. > > If "mueckenh" wants the string of the first three 1's as index for the > third 1, that is possible. > > > > > > > > If 1/9 is not in the list but > > > > can be indexed completely by list numbers > > Each digit in the decimal expansion can be indexed by a natural number > but the 'completed' expansion can only be "indexed" by the 'completed' > set of all naturals. Which is perfectly symmetric. > > > , this symmetry is broken, > > > > because: What means "not in the list"? It means that 1/9 has more 1's > > > > than every list number. > > It does not mean that 1/9 has a decimal expansion with "more" 1's than > there are naturals in the set of all naturals. All numbers with as many 1's as are indexible by naturals are in the true list. 0.111... is not in the true list. That is proof enough. > Regards, WM
From: Virgil on 8 Aug 2006 16:37
In article <1155068441.763266.291560(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > I am using pi but I know that it is not a number but only an idea. > False mathematics results only from the assumption that pi was a number > and that all of its digits could be enumerated by natural numbers. Then does "mueckenh" reject circles from geometry also? > There are self evident truths like 2 + 2 = 4, which do not require > fuzzy axioms. But in a field of three elements, 2 + 2 = 1. So without axioms (assumptions) and definitions up front, "2 + 2 = 4" is meaningless. When one starts with nothing, one ends with nothing. "Mueckenh" has his own assumptions but will not own up to them. > > > > Yup, and aleph-0 is *not* in the inductive set of natural numbers. > > It is not in the set, but the number of the elements of the set is > aleph_0. Under this aspect aleph_0 should have some existence in the > set, namely as the number of elements. Of course it doesn't. Therefore > it cannot be at all. Gobbledegook. Aleph_0 has a better existence that "mueckenh"'s non-mathematics. > That is obviously necessary, but it is wrong. In particular according > to Cantor who always speaks of the infinite set of *finite* numbers. > With regard to potential infinity that is correct, but with regard to > actual infinite it is wrong. But without actual infinity aleph_0 makes > no sense. There is no such thing as a merely potential infinity. > > 1) According to Cantor omega is a number which is in trichotomy with > other numbers and which is larger than any natural number. This is the > first assertion which I deny and disprove. "Mueckenh" can deny it but cannot disprove it without assuming the equivalent of its falsehood. > > 2) If we have only potential infinity Ain't no such animal unless one assumes its existence. "Mueckenh"'s secret assumptions are the point at issue. Until he makes them explicit, he cannot argue anything. > > When I talk about a natural number n Absent "mueckenh"'s axioms, he has no numbers or anything else. |