An uncountable countable set
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From: Virgil on 6 Aug 2006 15:00 In article <1154882695.990488.7420(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Sure it can, Just use the "Hilbert Hotel" method. rs. > > My list of all natural numbers is not a set in the pathological sense > of set theory. Then it is not a list of all naturals, as any list of all naturals requires a set of all naturals, or at least some inductive set, to index the list.
From: Virgil on 6 Aug 2006 15:02 In article <1154882787.391149.8240(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > 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. > > Only by assertion. Not by facts. By proofs, valid in ZF or NBG, which "mueckenh" must needs ignore, since he has no valid counters to them.
From: Virgil on 6 Aug 2006 15:04 In article <1154883048.729461.122990(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > But the *+ sum is defined for all digit positions which can be indexed > by a natural number. By "Hilbert Hotel" methods, one can always find room for another, or even countably many more.
From: Virgil on 6 Aug 2006 15:06 In article <1154883048.729461.122990(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > n is a natural number. An index is a natural number. "Index" and > "natural number" are synonymous. Not entirely. Any inductive set can be used for indexing, and not all of them are order isomorphic to N.
From: Virgil on 6 Aug 2006 15:22
In article <1154883863.552481.311990(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > Question: Which digit(s) of the decimal expansion of 1/9 cannot be > > indexed by some finite natural? > > Those by which 1/9 differs from any number in the true list What "true list" does "Mueckenh" have in mind that does not contain ALL the digits in the decimal expansion of 1/9? > > Answer: there is no digit in that expansion that cannot be indexed by a > > finite natural. > > > > To maintain the existence of such a digit requires proof that it exists, > > and "mueckenh" has no such proof because no such digit exists. > > It requires proof that 0.111.. does exist and can be indexed. But there > is no proof. The first "1" in "0.111..." can be indexed by {}. If any "1" can be indexed by x the following "1" can be indexed by (x U {x}) Conclusion: every "1" in "0.111..." can be indexed by an inductive set. |