An uncountable countable set
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From: Virgil on 28 Aug 2006 15:03 In article <1156765466.460748.268590(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1156500782.651084.171830(a)i3g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > In article <1156148773.420187.122140(a)h48g2000cwc.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > > > > You are dreaming. It does follow from the form of the numbers of my > > > > > list. > > > > > Try to come back to reality. Simply show how a digit position n can be > > > > > indexed the complete number of which, i.e., the complete sequence of > > > > > digit positions 1 to n is not in the list. > > > > > > > > The Hilbert Hotel method! > > > > > > Why does the Hilbert Hotel method not apply to my list, but only to > > > Dik's number? > > > > Because it DOES apply to your list! It applies to any list. > > Why then is 0.111... not in my list, but Hilbert helps only Dik to > defend his number being indexable? Because you are too willfully blind to see what others easily see.
From: Virgil on 28 Aug 2006 15:06 In article <1156765535.826927.97670(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1156500895.295710.85000(a)75g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > > > The set of edges of an infinite binary tree are certainly countable, and > > > > it is not difficult to construct an explicit bijection between them and > > > > the set of binary representations of the naturals. > > > > > > > > The set of paths, however is not countable, as may is shown by > > > > constructing an explicit bijection between the set of all such paths and > > > > the set of all subsets on the set of naturals. > > > > > > > > Both the bijections referred to above have been presented here with no > > > > one able to show either to be anything other than as advertised. > > > > > > Ok. Let's accept that. But by rational relation we find the set of > > > paths being *not* larger > > > than the set of edges. How is that possible? > > > > Because the "rational relation" in question requires that both be finite. > > That would contradict the axiom of infinity. How does existence of a relation between the set of edges and set of paths of finite binary trees contradict anything?
From: Virgil on 28 Aug 2006 15:13 In article <1156765673.054531.101830(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > The problem is not indexing but "indexing without covering". > > > That is easy to prove impossible for any finite natural number in unary > > > representation. And, alas, there are only finite natural numbers. > > > > I have no idea what "indexing without covering" means, > > Never mind, it is impossible. If it cannot be explained then it is irrelevant. A set is indexable by the naturals if there is a surjection from the naturals to that set, and one can construct a surjection , actually a bijection, from N to {0.1, 0.11, 0.111,...} \/ {0.111...}. > > But they HAD those specific definitions that are required. It is > > division in the absence of such definitions that I object to. > > They had the theory of proportion of geometric lines and areas. That's > all needed. In particular the division was (and is) not different for > naturals and fractions and reals. At least to the Greeks, division involving incommensurables was quite different from division involving only commeasurables.
From: Virgil on 28 Aug 2006 15:22 In article <1156765738.249634.122700(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1156501377.098380.50700(a)m79g2000cwm.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > > > > But as we just investigate consistency, you cannot presuppose it. With > > > > > your attitude it is impossible to find any inconsistency even in an > > > > > inconsistent theory. Deplorably you are too simple to recognize that. > > > > > > > > If "Mueckenh" can deduce from any axiom system both a statement within > > > > the system and its negation, "Mueckenh" will have found his > > > > inconsistency. > > > > > > Which part of my proof concerning the binary tree is not in accordance > > > with the ZFC axioms in your opinion? > > > > The part that says a bijective image of the naturals bijects with a > > bijective image of the power set of the naturals. > > I do not say that they biject. I have show several times explicit bijections. Does "Mueckenh"deny that any of them are actually as presented? > I only say that the bijective image of > the naturals has more elements (or more precisely: not less elements) > than the bijective image of the reals. If that were true then "Mueckenh" would be claiming existence of a surjection from the naturals to the reals. So to prove his claim, "Mueckenh" should be able to exhibit such a surjection. I will hold that he is wrong until "Mueckenh"has produced such a surjection from the set of naturals to the set of reals, as clear and explicit as my bijections mentioned above. > But that is only the consequence > which cannot be avoided if there are no other loopholes. I do not need loopholes for my bijections.
From: Virgil on 28 Aug 2006 15:27
In article <1156765804.338031.236870(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > But > whether the results describe reality too, is only a question of > mathematics being useful or useless. That is the view of an incurably non-mathematician. Mathematicians often prefer beauty to utility. Beauty, while useless to such as "Mueckenh", is not to be despised. |