An uncountable countable set
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From: Dik T. Winter on 25 Jul 2006 10:50 In article <1153830410.374836.199250(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > > I do not use the negation of the axiom, but accept that every > > > > > digit of 0.111... can be indexed by a finite natural. Hence > > > > > every sequence of digits of 0.111... can be covered by a natural. Note that you use "sequence" above, without qualification. > > > > You omit the "finite" between "every" and "sequence". > > > > > > If *every* digit can be indexed, then I need not put "finite" between > > > "every" and "sequence", because every indexing sequence is finite. Perhaps I misunderstood. What is your definition of indexing sequence? Perhaps you are meaning "sequence for which every element can be indexed by a natural number"? In that case your claim that every indexing sequence is finite is equivalent to every set of natural numbers is finite. This is clearly a contradiction of the axiom of infinity. > > Your deleting of "finite" (or "indexing") here is precisely what I said, > > the negation of the axiom of infinity. > > No. It makes use of the fact that every natural is finite. No, it is the negation of the axiom if infinity. Every natural is finite, but the set of naturals is infinite. > > Remember: there is a set that contains all > > naturals. There is (clearly) no natural that is the largest in that set. > > But everyone is finite. Yes, but the set is infinite. So with all naturals we can index an infinite sequence of digits. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Michael Stemper on 25 Jul 2006 12:52 In article <1153826418.132895.291750(a)h48g2000cwc.googlegroups.com>, mueckenh writes: >Franziska Neugebauer schrieb: >> > In set theory. It is full of the actual infinite. >> >> When I ask for "usage of terms" I want you to name contemporary >> papers/books on set thery that mention/use these terms. > >1976 is too old? Well, that book wasn't written in 1976; it was a reprint. Figure out when it was written, not when it was republished, and get back to us. > Few modern books spell it out because all think that >all know it. That's one possible interpretation. Another one is that modern books don't use the terms "actual infinity" and "potential infinity" because they aren't meaningful terms. >Is July 11, 2002 contemporary enough? Thomas Jech: Set Theory >Stanford.htm, Stanford Encyclopedia of Philosophy: Until then, no one >envisioned the possibility that infinities come in different sizes, and >moreover, mathematicians had no use for "actual infinity." Okay, so your own reference shows that philosophers don't think that mathematicians use the term "actual infinity". That sounds correct. >Dear Franziska, if you don't know that "infinity" in set theory always >means "actual infinity", I've never even encountered the term "infinity" in any of my (admittedly limited) studies of set theory. I've encountered the term "infinite", as an adjective preceding "set", but never just plain (and undefined) "infinity". -- Michael F. Stemper #include <Standard_Disclaimer> Reunite Gondwanaland!
From: Virgil on 25 Jul 2006 14:12 In article <1153825348.155179.195560(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > Mathematics cannot be done without matter. For thinking, speaking, > > > writing matter is required. Much matter allows much mathematics, less > > > matter allows less mathematics, no matter allows no mathematics. > > > > That is where we differ. To me axiomatic mathematics is entirely mental. > > And where does your thinking happen? In my mind, of course. That "mueckenh" does not have one can be the only reason he has to ask. > > > While the physical world has certainly suggested the directions in which > > mathematics has developed, those developments are themselves not a part > > of the physical world, but only of the mental world. > > > The patterns humans see in nature are imposed by the humans seeing them > > and are not inherent in nature itself. The observer is a part of what is > > observed. > > In particular he and his thinking is a part of reality. What one perceives as "reality" is only what filters through his thinking. What reality "is" and what people perceive it to be are not necessarily the same. And there are no numbers, nor geometric figures, nor any other mathematical notion outside thought.
From: Virgil on 25 Jul 2006 14:19 In article <1153825521.722967.109540(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > It is easy to show that the number of infinite paths is countable too, > > > by he fact that each path carries two edges as its load. > > > > > > If "mueckenh" claims it is so easy to show, I challenge "mueckenh" to > > establish a bijection, or even a surjection, from the set of naturals > > (or finite ordinals) to the set of all maximal paths in an infinite > > binary tree. > > | a > o > /b \c > > Map the first edge a on the bunch of those paths which have this edge > in common. Map the edges b and c by which the bunch splits in two > bunches on the corresponding bunches turning left and right, > respectively. And let one half of edge a be inherited by each of theses > bunches. Continuing we obtain exactly 2 edges for every bunch. But that does not give a different edge for each path, it only gives an edge for an infinite set (bunch) of paths. > > But sorry, I forgot, you hate new ideas like fractional mappings. (Call > it a relation if it helps you to understand.) What "mueckenh" seems to have forgot are the reqeuirements of the challenge. As whatever "mueckenh"'s "construction achieves, it does not achieve the goal of a unique and different edge for each separate path. It fails the challenge. As any attempt must always fail.
From: Virgil on 25 Jul 2006 14:21
In article <1153825676.234613.229870(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Here are few quots from many: Fraenkel, Abraham A., Levy, Azriel: > > > "Abstract Set Theory" (1976) > > > p. 6 the set of all integers is infinite (infinitely comprehensive) in > > > a sense which is "actual" (proper) and not "potential". > > > One may doubt whether this example really illustrates the abyss between > > > finiteness and actual infinity. > > > p. 240 Thus the conquest of actual infinity may be considered an > > > expansion of our scientific horizon no less revolutionary than the > > > Copernican system or than the theory of relativity, or even of quantum > > > and nuclear physics. > > > > > > Potential infinity existed already long before set theory. > > > > But actual infinity exists now, at least according to the quotes above. > > > > So that "mueckenh"'s own citations contradict his own thesis. > > > > Like an engineer hoist with his own petard! > > These are only quotes and definition, not proofs and truths. Quoting "mueckenh" to disprove "mueckenh"'s claims, is proof enough. |