An uncountable countable set
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From: Virgil on 6 Oct 2006 14:52 In article <1160122708.108138.77770(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > I am sure, the results "all balls in B" and "all balls not in B" are > not to be interpreted as an actual contradiction of set theory. It is > just counter intuitive. > > Regards, WM It seems to be intuitive in "Mueckenh" 's world, but it is not only not intuitive, it is not true in any set theory of my acquaintance.
From: Virgil on 6 Oct 2006 14:57 In article <1160123095.800410.99250(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160044514.105544.245260(a)c28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Dik T. Winter schrieb: > > > > > > > > > > > > > (There are exactly twice > > > > > > > so > > > > > > > much > > > > > > > natural numbers than even natural numbers.) > > > > > > > > > > > > By what definitions? You never state definitions. > > > > > > > > > > By the only meaningful and consistent definition: A n eps |N : > > > > > |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. > > > > > Do you challenge its truth? > > > > > > > > No, I never did. But you draw conclusions about it about the set N. > > > > Indeed, > > > > for each finite n, it is true. > > > > > > And N is nothing but the collection of all finite n. > > > > That does not require that what is true for every member of a set be > > true for the set itself. > > > > {2,4,6} is an odd sized set, despite all its members being of even size. > > And Mars looks red although all Marsians are green. Such analogies do > not prove anything. In particular a set of finite natural numbers > cannot be infinite, because the sum of differences of 1 between these > numbers also makes up a finite natural number, as long as only finite > numbers are present in the set. No one but idiots like you and TO claim that having infinitely many finite naturals requires having anything like an infinite natural. > But this sum is nothing than the number > of numbers (less 1). The "number" of naturals is not a natural. And a "sum" such as the one suggested, need not exist at all.
From: Virgil on 6 Oct 2006 14:58 In article <1160123274.697574.301020(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160066751.825020.117740(a)h48g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > MoeBlee schrieb: > > > > > > > A CONTRADICTION? From what AXIOIMS? Please show a derivation of a > > > > sentence P and ~P from the axioms. Oh, that's right, by "contradiction" > > > > you don't mean a contradiction in the sense of a sentence and its > > > > negation; you mean something that doesn't sit with your personal > > > > intuition. > > > > > > > > > > + 1 Gedankenexperiment: Put 10 balls in A and remove two, one of which > > > is put in B and the other one is put in C. > > > > > > P: At noon all balls are in B. > > > ~P: At noon all balls are in C. > > > > > > > > > Q: some are in each. > > Then let the volume of C be a subset of the volume of A. Now A being no > longer empty at noon? > > Regards, WM That would require putting balls that have been removed from A back into A, which is a different game.
From: Virgil on 6 Oct 2006 15:08 In article <1160124547.657300.325000(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > That axiom of infinity says, in symbols, there is an x such that {} is a > > member of x and FOR ALL y, if y is a member of x so is (y union {y}). > > That is a quantifyer which is not expressed as a word "all" meaning > that all are there. (In text versions we have only: if y is in x then > {y} is in x (Zermelo's version).) "FOR ALL" concerns all those y which > are in x but it does not state that a set of all y did exist. Because > then it would be easy to define this set by: "The set of all y". > > > > Regards, WM "if y is in x then {y} is in x" (Zemelo) (or "if y is in x then y union {y} is in x" ,Fraenkel) also requires that one consider y not in x. Or does "Mueckenh" assert that there are y for which "if y is in x then {y} is in x" is not relevant?
From: Virgil on 6 Oct 2006 15:09
In article <1160124783.798885.20040(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > > > > The axiom of infinity does only state n+1 exists if n is given. It is > > > > > realized by he numbers of my list. > > > > > > > > That is *not* what the axiom of infinity states. The axiom states that > > > > there exists a set that contains all the successors of its elements. > > > > > > Oh I must have read always wrong texts. I never came across the word > > > "all" in connection with this axiom. > > > > What text tells you that part of the axiom of infinity is that n+1 > > exists if n is given? > > If x belongs to the set then {x} belongs to it. From thia n --> n+1 > can be proved. > > > > And the quantifier 'for all' is part of the axiom of infinity: > > > > There exists an x such that 0 is a member of x and, for all y, if y is > > a member of x then yu{y} is a member of x. > > > > In symbols: > > > > Ex(0ex & Ay(yex -> yu{y]ex)) > > I know these symbols. A means "for all" y which are there, it does not > state that "all" y are there. > > Regards, WM How does "for all y which there are" differ from "for all y there are"? |