An uncountable countable set
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From: Tony Orlow on 8 Oct 2006 15:12 Lester Zick wrote: > On Sat, 07 Oct 2006 23:45:23 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> mueckenh(a)rz.fh-augsburg.de wrote: >>> Tony Orlow schrieb: >>> >>>> mueckenh(a)rz.fh-augsburg.de wrote: >>>>> Tony Orlow schrieb: >>>>> >>>>> >>>>>>>> Why not? Each and every number of the list terminates. That one is a number >>>>>>>> that does *not* terminate. >>>>>>>> >>>>>>>> > If you think that 0.111... is a number, but not in the list, >>>>>>> It is me who insists that it is not a representation of a number. >>>>>> Well, Wolfgang, that sets us apart, though I agree it's not a "specific" >>>>>> number. It's still some kind of quantitative expression, even if it's >>>>>> unbounded. Would you agree that ...333>...111, given a digital number >>>>>> system where 3>1? >>>>> That is the similar to 0.333... > 0.111.... But all these >>>>> representations exist only potentially, in my opinion. The difference >>>>> is, that 0.333... can be shown to lie between two existing numbers, so >>>>> we can calculate with it, while for ...333 this cannot be shown. >>>> I think it can be shown to lie between ...111 and ...555, given that >>>> each digit is greater than the corresponding digit in the first, and >>>> less than the corresponding digit in the second. >>> Yes, but only if we define, for instance, >>> >>> A n eps |N : 111...1 < 333...3 where n digits are symbolized in both >>> cases. >>> >>> This approach would be comparable with the "measure" which gives >>> >>> A n eps |N : |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. >>> >>> I don't know whether these definitions are of any use, but I am sure >>> that they are not less useful than Cantor's cardinality. >>> >>> Regards, WM >>> >>> . >>> >> My opinion about that is, if one wants to talk about what happens "at >> infinity", that's the way that makes sense, not the measureless way of >> abstract set theory. I trust limit concepts, but not limit ordinals. > > Tony, would it be fair to characterize what you're trying to say as > that there is some kind of positive/negative crossover at infinity > such that {-00, . . .,-1, 0, +1, . . . +00}? I haven't really been > following this thread too closely so I'm trying to understand what > you're after here in basic terms instead of the exact arguments > involved. > > ~v~~ Hi Lester, how's thangs? I wasn't saying that right here, but agreeing with Wolfgang that limit concepts make sense, while the transfinitological approach doesn't. What I did say was that there are two ways to view the number line, one where oo and -oo are polar opposites, and a number circle where they are the same. When it comes to reality, pretty much everything is in circles, including finite number systems so this model makes some sense, even if it doesn't in terms of limits of, say, powers. lim(x->-oo)=0 and lim(x->oo)=oo for n^x where n>1, so there the two are different. When it comes down to this argument, Wolfgang's argument, I agree with his logic concerning the naturals and the identity function between element count and value. He chooses then to reject infinities, while I choose to retool them. I think the problem he and I both see clearly is that using finiteness as a "bound" on the set simply doesn't tell you anything, but rather clouds the entire subject. Did that answer your question? :) Tony
From: Virgil on 8 Oct 2006 15:19 In article <1160309417.961906.163060(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. But "if x eps Z, also {x} eps Z" means "For all x, if x eps Z, also {x} eps Z" if it is to mean anything at all.
From: Virgil on 8 Oct 2006 15:23 In article <1160309564.676537.186720(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. 3) there is no bijection between the cardianal of the number of balls in the vase after all escapes and any set larger than {}. > > (Instead of "balls", use "elements of X where X is a variable".) Unless X is a set, "elements of X" is nonsense. > Regards, WM
From: Virgil on 8 Oct 2006 15:27 In article <1160309951.808867.33730(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > 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. To claim that the number of naturals must be a natural is to require indiscriminately that sets be allowed to be members of themselves, which causes worse logical problems than any "actual" infinities have ever caused.
From: Virgil on 8 Oct 2006 15:30
In article <1160310024.534838.300800(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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 Then why not stay just away form it. Those of us who find ZF, ZFC and NBG, etc., worth considering find the continual preaching of those dogmatists who do not offensive. |