An uncountable countable set
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From: MoeBlee on 16 Oct 2006 17:34 Han.deBruijn(a)DTO.TUDelft.NL wrote: > MoeBlee schreef: > > > Han de Bruijn wrote: > > > Virgil wrote: > > > > Axiom of infinity: There exists a set x such that the empty set is a > > > > member of x and whenever y is in x, so is S(y). > > > > > > Which is actually the construction of the ordinals. Right? > > > > Wrong. > > Don't understand why that's wrong. Please explain. It's not the definition of 'ordinal' and there are ordinals that are not "constructed" or proven to exist by the axiom of infinity. The axiom of infinity is just not the construction of the ordinals. Why don't you just read a set theory textbook rather than remain ignorant about that which you are so opinionated. MoeBlee
From: MoeBlee on 16 Oct 2006 17:37 Han de Bruijn wrote: > Virgil wrote: > > > In article <1160935613.121858.178420(a)m7g2000cwm.googlegroups.com>, > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > >> I find the idea absurd that natural numbers can be built > >>by putting curly braces around the empty set. > > > > It appears as if much of useful mathematics is ultimately based on what > > HdB finds absurd. > > The natural numbers can be defined without employing set theory. Yeah, so? And, by the way, natural numbers are not defined in set theory by a method of curly braces. MoeBlee
From: MoeBlee on 16 Oct 2006 17:45 David Marcus wrote: > Ross A. Finlayson wrote: > > There is no universe in ZF, ZF is inconsistent. > > What exactly is the inconsistency, please? Of course he'll never show you an inconsistency. "Set theory is inconsistent" is just one among the phrases that he's fond of muttering. MoeBlee
From: Virgil on 16 Oct 2006 17:57 In article <4533d315(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45319e2d(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <453108b5(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>>> So how many balls are left in the vase at 1:00pm? > >>>>> > >>>> If you paid attention to the various subthreads, you'd know I just > >>>> answered that. Where the insertions and removals are so decoupled, there > >>>> is no problem. Where the removal of a ball is immediately preceded and > >>>> succeeded by insertions of 10, the vase never empties. > >>> It may that it never "empties", but at noon, and thereafter, it has > >>> "become empty". > >>> > >> And when did that happen? > > > > At the invisible transition from forenoon to noon. > > Oh, you mean the moment between all moments before noon and the moment > of noon? No! The set of all moments before noon is, in the real number model of times, an open set whose least upper bound, noon, is not a member of that set but such that there are no times between that set and its LUB, noon. So that as sets of times, the forenoon and what comes after, form a partition of times into disjoint sets with every time being in or or the other but none in both. > > >>> The only necessary constraint on insertions of balls into the vase and > >>> removals of balls from the vase is that each ball that is to be removed > >>> must be inserted before it can be removed, and, subject only to that > >>> constraint, the set of balls remaining in the vase at the end of all > >>> removals is independent of both the times of insertion and of the times > >>> of removal. > >>> > >>> To argue otherwise is to misrepresent the problem. > >> You already said that.....WRONG!!!! > > > > What is wrong about it? > > This: > > >> There is the additional constraint that, before removing any ball, ten > >> have been inserted. > > > > Then let us put all the balls in at once before the first is removed and > > then remove them according to the original time schedule. > > Great! You changed the problem and got a different conclusion. How > very....like you. Does TO claim that putting balls in earlier but taking them out as in the original will result in fewer balls at the end? If so, by what logic, and if not, what difference does it make? > > > > > Does TO claim that by putting balls in earlier there can be at ANY time > > fewer balls in the vase that when putting them in later? > > Yes, I've already explained that. Not to anyone else's satisfaction. > > > > > But when one puts them all in early enough, it becomes obvious that the > > vase must be empty at noon. > > Yes, if all insertions occur before all removals. How does that change things? Is there any time at which putting balls in earlier forces fewer balls to be in the vase? If so, at what time does the number of balls from the earlier insertions become less that the number of balls from the later insertions. As far as I can see, putting balls in earlier can only increase the number of balls in the vase at some times, but not decrease the number at any time. > > > > So TO must argue that having more balls in the vase at all times before > > noon results in less balls in the vase at noon. > > > > Now that is ...REALLY... WRONG!!! > > > > > > It is correct to say that decoupling the insertions from the removals > such that each sequence is distinct time-wise and with its own point of > condensation creates a different situation. As long as each ball is inserted at least as early as originally and no ball is removed earlier that originally, if there is to be any set of times at which there are fewer balls that originally then it must be after the first ball is inserted in the new schedule, so the times are bounded below and must have a GLB (greatest lower bound). What is that GLB? Unless you can provide one, your claim fails.
From: MoeBlee on 16 Oct 2006 18:05
Tony Orlow wrote: > No, set theory confuses the issue with its concentration on omega. Oh boy, here we go again with "Set theory confuses...". Please just way which axioms of set theory you reject and which ones you use instead. MoeBlee |