An uncountable countable set
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From: mueckenh on 25 Jun 2006 05:16 Virgil schrieb: > > There is no mapping including K as the image of at least one source > > element, even if surjectivity is not at all in question! > > That is not the issue in question. > "Mueckenh" has claimed that he has produced an example of a set that is > both countable and uncountable (See Subject), but his example is flawed > because the sets and function he has presented do not exist as he has > presented them. I made this claim well aware that you would substitute f by another mapping g. This should show you that Cantor's diagonal method can also be treated in the same way showing that it doesn't show anything about uncountability. > Regards, WM
From: mueckenh on 25 Jun 2006 05:19 Virgil schrieb: > In article <1151154151.237811.168860(a)y41g2000cwy.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > Actually, even if it IS true, it would not allow transmutation of a set > > > with a first element into one without a first element, at least if the > > > transpositions are applied sequentially, as they must be. > > > > The transpositions are just as sequentially as the construction of the > > diagonal number. > > > > > There would have to be a first transposition in the sequence which > > > effects the change, and a single transposition is incapable of doing > > > this. > > > > This arguing stems from König (1905). We know that there are only > > countably many constructible reals. If there is a well-ordering of the > > reals, then within this order there must be a first non-constructible > > real. This is a paradox. Hence, there is no well order. > > No one here is claiming a well-ordering of the reals. You don't accept ZFC??? Regards, WM
From: mueckenh on 25 Jun 2006 05:22 mueckenh(a)rz.fh-augsburg.de schrieb: > Virgil schrieb: > > > > If an almighty God cold not arrange to have the sun appear to stand > > still in the sky without disastrous side effects, such a God would > > hardly be almighty. > > Of course not. He would not even be able to construct a stone so heavy > that he was unable to lift it. > > If such a God first says: It is not good for the man to stay alone, so > we will create a women for him. Oh, HE created only one woman, of course. Regrads, WM
From: Daryl McCullough on 25 Jun 2006 09:41 mueckenh(a)rz.fh-augsburg.de says... >I made this claim well aware that you would substitute f by another >mapping g. This should show you that Cantor's diagonal method can also >be treated in the same way showing that it doesn't show anything about >uncountability. Well "P(N) is uncountable" means "there is no surjection from N to P(N)", which means "for every function f from N to P(N), f is not a surjection", which means "for every function f from N to P(N), there is an element K of P(N) such that K is not in the image of f". Since K(f) is an element of P(N), and K(f) is not in the image of f, then yes, this does show something about uncountability. -- Daryl McCullough Ithaca, NY
From: Virgil on 25 Jun 2006 13:42
In article <1151224036.572450.117050(a)y41g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Daryl McCullough schrieb: > > > mueckenh(a)rz.fh-augsburg.de says... > > > > >"Surjective" means that all elements are in the image. A potentially > > >infinite set is always finite though surpassing any given limit. > > >Therefore the question of all of them cannot be decided. > > > > That's not true. If A is a "potentially infinite" set, then > > saying "forall x in A, Phi(x)" just means that if x is any > > thing of type A, then Phi holds of x. It doesn't matter whether > > A is "completed" or not. > > > > We know perfectly well that if x is a natural number, then x+x > > is an even natural number. From this fact, it follows that > > > > forall x: Natural, x+x is even > > > > "forall" doesn't mean that anyone has *checked* every element of > > the set. > > But it doesn't say what "all" means. "For all x f(x)" means "there is no x for which f(x) does not hold" >The set hasn't a fixed > cardinality, because for every n considered, there is an n+1 not yet > considered, because "potential" means always finite but not limited by > any threshold. If you can show that there is no 'n' for which the property does not hold, you have simultaneously shown that it holds for all 'n'. Something is true in all cases if and only if there are no exceptions. We are never closed. A diagonal number is never ready. The "diagonal number" of Cantor's 2nd proof is established by a rule which allows of no exceptions. > The set K is never ready, unless we chose it to be finite by a suitable > f. The function f is never "ready" unless we require that K not be in it image. |