An uncountable countable set
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From: Virgil on 15 Sep 2006 14:41 In article <1158311237.866137.121980(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > III is a representation of 3. > > > > "III" and "3" are both numerals representing the same number, but > > neither is anything more than a representation, neither is the number > > itself. > > What is the number denoted by 3? Some cloudy idea in the platonic > universe or heaven? The number 3 is present in this very line. It is > "3" and it is "III" and it is "{a,b,c}". Everything else is purest > matheological rubbish. > > Regards, WM "Mueckenh" has now descended to multiple postings of his more egregious errors.
From: Virgil on 15 Sep 2006 14:44 In article <1158311345.901825.271970(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > Counting can be done by making tally marks or moving pebbles, for > > example, entirely without numbers, though we have become so > > sophisticated that we may have trouble realizing it. > > > > It is whether the tally marks or collected pebbles biject with the > > objects counted which is the issue. > > > > And no number need ever be mentioned or used. > > Your tally marks and moving pebbles are numbers. Get more sophisticated > in order to see it. Numbers derive from tally marks and pebbles as common properties, but neither tally marks nor pebbles ARE numbers, or even signify numbers, without a mind to interpret them, so numbers exist only in the mind.
From: Virgil on 15 Sep 2006 14:53 In article <1158311704.687685.49680(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Does the path of 0.111... terminate? Does the path of pi terminate? > Each of their edges terminates. Pi does not have any edges. Approximations for pi have digits. > > > > > > This example says: Countability can be proven without a bijection. > > > > A misapprehention. Countability is *by definition* the presence of a > > bijection with (a subset) of the set of natural numbers. > > Aleph_0 is defined by the presence of a bijection with N. But if people > assert that aleph_0 is larger than any natural number and smaller than > 2^aleph_0, i.e., that aleph_0 has properties of a number, then one can > test that property with more general means like the rational relation. > This unavoidably and unambiguously leads to the result that aleph_0 is > the number of edges is not less than the number of paths is 2^aleph_0. Garbage. A set having cardinality aleph_0 will not inject into any set having finite cardinality but every set having finite cardinality will inject into it. A set having cardinality equal to the power set of the naturals will not inject into any set having cardinality aleph_0 but every set having cardinality aleph_0 will inject into it.
From: Virgil on 15 Sep 2006 14:58 In article <1158311965.841357.58540(a)d34g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Cantor's truths are self-evident truths which cannot be changed > arbitrarily in contrast to the axioms of modern set theory. Cantor's "truths" are as much based on assumptions as any other "truths" in mathematics, and ultimately those assumptions cannot be prove true. Those who claim self-evident truths in mathematics deceive themselves. Only those who can clearly state what they are assuming, as in a set of axioms, can be free from such self-deceits.
From: Virgil on 15 Sep 2006 15:07
In article <1158312563.101733.242910(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > Yes, it is not in the list. So what is the problem? All possible > > > > indexes > > > > are in the list, so: > > > > (1) A{p = digit position} E{q = list item} {such that q indexes > > > > p} > > > > which you deny. > > > > > > That is not an argument. > > > > *Why* is it not an argument? You state: "all possible indexes are in the > > list". What is *wrong* about the argument? You always state "it is > > wrong", > > but never state what part of the argument is wrong. > > It is wrong because not for all {p = digit position} there exists {q = > list item} {such that q indexes p} , but only for those p, which can > be indexed, i.e., which are present in the list. As 0.111... is not in > the list, there must be a digit position, which is not in the list. (As > such a position cannot be identified, 0.111... cannot exist.) Again "Mueckenh" claims, but in proof merely reiterates his claim without proving it. And as others have demonstrated that his claim is spurious, by indexing every digit in "0.111..." along with "0.111..." itself, "Mueckenh"'s disproved claim is false. > > > > > But there is a single "number" that can be indexed by your list but not > > covered by your list. 0.111... is one (as I defined it above). But it > > is *not* a natural number. > > Therefore it cannot be indexed by natural numbers. The non-numbers in {a,b,s} clearly can be indexed by natural numbers, so what prevents any single object being so indexed? > Try to find out, why > you insist on only one of those strange numbers. There could be > infinitely many. Could they all be indexed by natural numbers? Any countable set of them could be, as that is what countability requires. > > > > Do you agree that the non-terminating list can index non-terminating > > numbers? If the answer is no, why? > > Because it should index at least two different of those infinite > numbers if it could index one of them. Since the naturals can index a countable infinity of them , indexing a mere two of them is no problem. |