An uncountable countable set
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From: mueckenh on 25 Jul 2006 07:02 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? > 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. Regards, WM
From: mueckenh on 25 Jul 2006 07:05 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 sorry, I forgot, you hate new ideas like fractional mappings. (Call it a relation if it helps you to understand.) Regards, WM
From: mueckenh on 25 Jul 2006 07:07 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. Are you so limited or do you simulate it only (but with excellence)? Regards, WM
From: mueckenh on 25 Jul 2006 07:09 Randy Poe schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Randy Poe schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Of course. Real space is Euclidean or non-Euclidian irrespective of the > > > > axioms which mathematicians prefer. Aithmetic is finite or infinite > > > > irrespective of the axioms which mathematicians prefer. > > > > > > You mean there's something called "arithmetic" which > > > exists outside mathematics? > > > > It is the core of mathematics. > > So the core of mathematics is outside mathematics? No, but present "mathematics" has little to do with real mathematics. Regards, WM
From: mueckenh on 25 Jul 2006 07:20
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? Few modern books spell it out because all think that all know it. 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." The arguments using infinity, including the Differential Calculus of Newton and Leibniz, do not require the use of infinite sets ... > > Set theory is theory of the transfinite. Transfinite is actually > > beyond the finite. > > > > Potentially infinite means always finite but without border. > > I don't ask for an elucidation of the terms but of accepted and > widespread use in current literature. Dear Franziska, if you don't know that "infinity" in set theory always means "actual infinity", then, I am afraid, I can't help you as little as with the use of logic. > > >> Where is the mathematical use of the aforementioned terms? > > > > In set theory! > > You will certainly find means in the world wide web which enable you to > substanciate your claim by giving us statistical evidence. But I will not search for them because I know the facts. (Therefore I don't know the address of Jech's quote above. I have taken it from the memory of my computer.) Regards, WM |