Cantor Confusion
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From: Virgil on 29 Oct 2006 19:09 In article <1162156965.871089.191790(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1162068521.341383.99300(a)k70g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > And once again WM deliberately confuses, "all positions are > > > > finite", with "there are a finite number of positions". > > > > > > There is no confusing! Every finite position belongs to a finite > > > segment of positions (indexes). If you don't believe that and assert > > > the contrary, then try to find a finite position which dos not belong > > > to a finite segment (of indexes). > > > > It is equally true that for every finite segment of indexes there are > > positions beyond that segment. > > Correct. And those positions have finite indexes too. > Every set of natural numbers has a superset of natural numbers which is > finite. Every! Not so. The set of even natural numbers has no finite superset. The set of prime natural numbers has no finite superset. > > > It is simply purest nonsense, to believe that "all positions are > > > finite" if "there are an infinite number of positions". > > > > There are certainly more than any finite number of positions. > > And it is even purer nonsense to believe that in such a situation there > > are only finitely many of them. > > Try to draw the graph of the function f(n) = n. The points lay on the > diagonal of the first quadrant. As long as n is finite, f(n) is finite > too and vice versa. So what.
From: Virgil on 29 Oct 2006 19:13 In article <1162157106.286220.176540(a)e64g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1162069321.729271.190360(a)f16g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > > > Correct. And therefore no such thing can exist unless it exists > > > > > > > in the > > > > > > > mind. But we know that there is no well order of the reals. in > > > > > > > any > > > > > > > mind, because it is proven non-definable. > > > > > > > > > > > > It is perfectly definable, and perfecty defined, it is merely > > > > > > incapable > > > > > > of being instanciated. > > > > > > > > > > Then let me know one of the perfect definitions, please. > > > > > > > > A set is well ordered when every nonempty subset has a first member. > > > > > > Which is the first member of the subset of positive reals? > > > > Under what ordering? if you claim an instanciation of such a well > > ordering, present it. I do not. > > What kind of well ordering of the reals do you claim to exist? A well ordering in which each non-empty subset has a smallest member. > Defined? > Catalogue? > List? > Else? (Please specify) A well ordering in which each non-empty subset has a smallest member. That's what well orderings are all about. If you are asking for a rule for determining which objects come before others, you should know that no such explicit rule is possible, but that does not make their existence imposible.
From: Virgil on 29 Oct 2006 19:22 In article <1162157208.320679.159860(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1162069566.039416.155090(a)k70g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Any real number has only finite digit positions, according to Dik > > > > > > > > And is Dik your "Authority"? > > > > > > No, but his answer shows that your party contradicts your own opinion. > > > > How do you deduce that because Dik and I agree that you are wrong that > > Dik and I are necessarily agree on everything or are "of the same party"? > > > > This is not a two party system. > > No? According to my impression, it is. Then WM's impressions are wrong, just as his convictions are.
From: Virgil on 29 Oct 2006 19:32 In article <1162157584.352015.276510(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > Euclides was apparently smarter than Cantor in this. He used the > > parallel axiom (postulate) for something he could not prove from the > > other axioms. > > But which is obviously possible and correct under special > circumstances, while Zermelo's AC is obviously false under any > circumstances. That obviousness is less than obvious to most of us. There are no circumstances that are so obviously true that they make ZF or AC obviously false. > > > > But when mathematicians are bothered about the parallel axiom have > > problems with the justification of that axiom and reject it, you > > aparently have no problems. But when mathematicians do the same with > > the (implicit) well-ordering axiom from Cantor you appear to have > > problems. Why? > > Because I can construct parallels (or see their absence) but I cannot > see a well-order which is not definable. WM's limitations are only his own, and need not limit others. > > > > With the axiom of choice, it is possible to well-order the reals. > > But it can also be shown that such a well-ordering can not be defined > > by a formula in ZFC. > > How then can the well-ordering be accomplished? Zermelo proved tat it > could be accomplished. > > Regards, WM
From: Ross A. Finlayson on 29 Oct 2006 19:37
Virgil wrote: .... > > A well ordering in which each non-empty subset has a smallest member. > > That's what well orderings are all about. > > If you are asking for a rule for determining which objects come before > others, you should know that no such explicit rule is possible, ...., where V =/= L, ... > but that does not make their existence imposible. In bijecting the reals to an ordinal, i.e., well ordering them, you might find it of interest to consider why Cantor's first, nested intervals, applies, or as you would say doesn't. As you won't use the natural ordering, construct the sequences of endpoints, to the degenerate interval. Set? Skolemize: countable. Some time ago, last year or so, we were discussing well-ordering the reals and where various attempts to well-order the reals go in their consideration. One notion is that if Not CH then non-measurable sets exist, and vice versa. That led to questions about measure, and fundamental results in measure theory that lead to non-measurable sets, eg Vitali, Banach-Tarski. http://groups.google.com/group/sci.math/browse_thread/thread/62eed929fe7c2ffe/6b053be28dac3662 Ross |