An uncountable countable set
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From: Virgil on 13 Aug 2006 14:02 In article <1155486885.745105.193680(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1155314988.698343.137140(a)74g2000cwt.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > In article <1155067333.259873.193700(a)p79g2000cwp.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > > > The v. Neumann model of natural numbers is just the model of > > > > > segments. > > > > > n = {0, 1, 2, ..., n-1} > > > > > Every natural number is finite. > > > > > Hence every union of finite segments is a finite segment. > > > > > > > > Every finite union of finite segments is a finite segment. Your hence > > > > does not follow. Again, a misinterpretation. > > > > > > Every union of finite segments is a finite segment. "finite" does not > > > apply here because there are *infinitely* many finite numbers. > > > > Just because there are infinitely many finite numbers, the word "finite" > > *does* apply. An infinite union of finite segments is *not* necessarily > > finite. Where do you find that an infinite union of finite segments is > > finite? Prove it. > > By definition each natural number is finite. As v. Neumann's model > gives every natural number but nothing else, every member of the set is > finite. There are infinitely many finite naturals in the von Neumann model, as the union of them all is an infinite set. > You keep on intermingling number of numbers and sizes of numbers. > Every number of the infinitely many numbers is a finite number. That > alone is important. But the set of all of them is not finite. that is or equal importance in NBG.
From: Virgil on 13 Aug 2006 14:07 In article <1155487078.836213.291820(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155314675.845888.174190(a)i3g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > So we have infinitely many finite triangles, but without that "final > > > > edge", no infinite triangle. > > > > > > Cantor's list has no final line. Is it finite? > > > > Since a triangle requires 3 edges, without all 3 it is not a triangle. > > An endless list does not require an end, so is "complete" without one. > > A symmetric rectangular triangle is completely determined by one edge > next to the right angle. But edges, as sides of triangles, being segments, must have two ends, and that "edge" does not.
From: Virgil on 13 Aug 2006 14:14 In article <1155487181.028039.253190(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155314605.714007.167990(a)i3g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > The line number is infinite by the axiom of infinity. > > > > Where in the axiom of infinity are infinite number lines mentioned? > > > > > If no line is numbered as omega, where can we find the line omega + 1? > > > > Who says there is one? > > Set theory. Where in set theory are any "lines" mentioned? > > > > > > > > Thus "mueckenh" is claiming to be able to inject the power set of the > > > > naturals into the set of naturals. > > > > > > > > We should be interested in seeing his attempts to perform this > > > > impossibility. > > > > > > Where does *my arguing* fail? It assumes without proof that what holds in finite cases must hold in an infinite case. > > What is > > true only for finite cases need not be true for infinite ones. > > What is true up to every level n of the tree is true for the whole > tree. Then "Mueckenh" claims the infinite tree must be finite as every finite level of it is finite. > > Else: What is true for a finite segment of Cantor's list need not be > true for the whole infinite list. The Cantor statement is not about finite segments of the list but about individual members of the list, that the constructed number is not equal to the number in place n in place n. > > What is the difference (in concluding from finity to infinity) between > list and tree? Lists don't branch.
From: Virgil on 13 Aug 2006 14:25 In article <1155487327.481835.100550(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155314314.212493.69980(a)75g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > > > The "Hilbert Hotel" method allows one always to insert one more and > > > > still have rooms for all. > > > > > > Just this argument shows that the diagonal number of Cantor's list has > > > always reserved a place in the list. > > > > Wrong. It may have a place in a new list, but none in the original list. > > > > What Cantor does say that is that no list can be complete, but does not > > say that there is any number which cannot be listed. > > We know that the set of *all* those numbers which can ever appear in > lists, be it as original entries or as diagonal numbers, is countable. > Even you know it! Why do conclude from Cantor's idea such incoherent > nonsense? Cantor never gets into the issue of whether there are inaccessible numbers. That issue did not come up until later. He merely says that no list can include all real numbers. Which is true. > > Regards, WM
From: Franziska Neugebauer on 14 Aug 2006 06:56
mueckenh(a)rz.fh-augsburg.de wrote: > An infinite sum of 1's is not infinite? n lim sum 1 = lim n =def L n -> oo i = 1 n -> oo There is no such L in N. F. N. -- xyz |