An uncountable countable set
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From: mueckenh on 8 Oct 2006 08:10 MoeBlee schrieb: > The point is that the quantifier 'for all', just as that quantifier is > understood throughout mathematics, does appear in the axiom of > infinity. Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: "Foundations of Set Theory", 2nd edn., North Holland, Amsterdam (1984), p. 46: AXIOM OF INFINITY Vla There exists at least one set Z with the following properties: (i) O eps Z (ii) if x eps Z, also {x} eps Z. There are several verbal formulations dispersed over the literature without any "all". In German: Unendlichkeitsaxiom: Es gibt eine Menge, die die leere Menge enthält, und wenn sie die Menge A enthält, so enthält sie auch die Menge A U {A} (oder die Menge {A}). Regards, WM
From: mueckenh on 8 Oct 2006 08:11 Virgil schrieb: > In article <ac6c7$45260f70$82a1e228$27946(a)news2.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > > Sure. Simply "define" something that's undefined. And create the self > > fulfilling prophecy that suits you best. > > Beats hell out of defining things that are already defined. > > Every definition worth having defines something that would be undefined > without that definition. A definition is an abbreviation. Regards, WM
From: mueckenh on 8 Oct 2006 08:12 Virgil schrieb: > 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. 0) There is a bijection between the set of balls entering the vase and |N. 1) There is a bijection between the set of escaped balls and |N. 2) There is a bijection between (the cardinal numbers of the sets of balls remaining in the vase after an escape)/9 and |N. (Instead of "balls", use "elements of X where X is a variable".) Regards, WM
From: mueckenh on 8 Oct 2006 08:19 Virgil schrieb: > > No one but idiots Your behaviour drops below the acceptable level. If you cannot recover, I must cease this discussion. > > 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. Every natural is the sum of 1 + as many 1 at it has predecessors. An infinite sequence of predecessors gives an infinite result. If no infinite sequence of predecessors exists, then only a finite sequence does exist. Regards, WM
From: mueckenh on 8 Oct 2006 08:20
Lester Zick schrieb: > On Fri, 06 Oct 2006 12:52:43 -0600, Virgil <virgil(a)comcast.net> wrote: > > >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. > > So it's counter intuitive. What's the problem? Surely it isn't the > first counter intuitive suggestion you've ever run across. I did not know that the simultaneous existence of A and ~A is not a contradiction but only counter intuitive. Now I learned it, and I find ZFC is not very attractive to me. Regards, WM |