An uncountable countable set
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From: Han.deBruijn on 1 Oct 2006 14:30 Randy Poe wrote: > Tony Orlow wrote: > > Randy Poe wrote: > > > Tony Orlow wrote: > > >> Han de Bruijn wrote: > > >>> Virgil wrote: > > >>> > > >>>> In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, > > >>>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > >>>> > > >>>>> Randy Poe wrote, about the Balls in a Vase problem: > > >>>>> > > >>>>>> It definitely empties, since every ball you put in is > > >>>>>> later taken out. > > >>>>> And _that_ individual calls himself a physicist? > > >>>> Does Han claim that there is any ball put in that is not taken out? > > >>> Nonsense question. Noon doesn't exist in this problem. > > >>> > > >> That's the question I am trying to pin down. If noon exists, that's when > > >> the vase supposedly empties, > > > > > > Why does the existence of noon imply there is a time > > > which is the last time before noon? > > > > > > It doesn't. > > > > I never said it did. When did I say that? > > I was responding to Han, who said that "If noon exists, that's when > the vase empties". HdB never said such a thing. > Noon exists. Sure. By dogma. Randy Pope is infallible. > But in order for the vase to transition from not-empty > to empty, there would have to be a last non-empty > moment. That would be the last time before noon. Aha. As clear as a klontje. > > I will offer this simple > > logical argument. If the vase ever became empty, it would be because one > > ball was removed, > > Hence my continued statement that the vase does not > "become empty". It is non-empty at certain times and > empty at others. There is no transitional moment. Nature does not jump, Leibnitz said. > Noon is the first moment at which the vase is empty. > But noon is not the transitional moment. There's no > time just before noon where the transition happened. Wow ! And _that_ calls himself a physicist ... Han de Bruijn
From: Virgil on 1 Oct 2006 14:38 In article <1159711218.812268.276490(a)c28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1159648393.632462.253170(a)k70g2000cwa.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > (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? I challenge the "truth" of its being the ONLY meaningful and consistent definition. "Mueckenh"'s claim is like that of a blind man claiming that colors are imaginary. I, and many others, find both meaning and consistency in the definition of cardinality. That "Mueckenh" does not is more a measure of his incapacity than of any lack of meaning and consistency. > > > Take into account that also Cantor's diagonal argument cannot be > > > executed in unary representation. > > > > Two red herrings in a single sentence. Can you get more? > > (1) Cantor's diagonal argument was about countable sequences of two > > symbols. There is only one countable sequence of one symbol. > > Cantor's argument was about reals. He strived for generality but did > not see that two symbols are not enough. > > > (2) Cantor's argument as augmented by Zorn and later by somebody who > > I do not know can not be executed for reals represented in base > > 3 or smaller. But reals are not tied to their representation. > > Therefore a general truth should not depend on the base 4 or larger. A specific proof of a general truth can be based on whatever it is based on. There are other proofs , including Cantor's first proof, which do not depend on any sort of representations of the reals. > > > You have not *shown* that, but defined it, erroneously. But if you had > > > shown it, then 0.111... was in the list, which also would have been > > > wrong. > > > > Stated without proof at all. What is erroneous about my definition? > > Do you assert that definitions can be erroneous? If so, why? Do you > > think the definition > > Let a be the number such that a = 4 and a = 5 > > is erroneous? I think not. It is a proper definition, but there is just > > no 'a' that satisfies the definition. > > It is erroneous, because you say let a *be* which is false, if a cannot > *be*. There is no such thing as an "erroneous" definition, except possibly in the sense of a grammatically incorrect one. A definition may lack any instantiation, such as a 4 sided triangle, but as a definition is valid.
From: Virgil on 1 Oct 2006 14:43 In article <1159725088.430659.72630(a)m7g2000cwm.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > stephen(a)nomail.com wrote: > You have fundamentally changed the "experiment". > > Worse. I have fundamentally changed the mathematics. > > Han de Bruijn HdB only has the power to dictate his own version of mathematics. He has not the power to control anyone else's mathematics.
From: Han.deBruijn on 1 Oct 2006 14:56 Virgil wrote: > In article <1159725088.430659.72630(a)m7g2000cwm.googlegroups.com>, > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > stephen(a)nomail.com wrote: > > > You have fundamentally changed the "experiment". > > > > Worse. I have fundamentally changed the mathematics. > > > > Han de Bruijn > > HdB only has the power to dictate his own version of mathematics. He has > not the power to control anyone else's mathematics. How do you know I'm not teaching mathematics to anyone? Han de Bruijn
From: Tony Orlow on 1 Oct 2006 15:14
imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> imaginatorium(a)despammed.com wrote: >>> Randy Poe wrote: >>>> Tony Orlow wrote: >>>>> Virgil wrote: >>>>>> In article <451bac34(a)news2.lightlink.com>, >>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> >>>>>>>>> If the vase is empty at noon, but not before, how can that not be the >>>>>>>>> moment that it becomes empty? >>>>>>>> Saying that it is empty is quite different from saying anything about a >>>>>>>> "last ball". andy does not deny that the vase becomes empty, he just >>>>>>>> does not say anything about any "last ball out". >>>>>>> Does that answer the question of **when** this occurs? Of course not. >>>>>> It does answer the question of "whether" it occurs. "When" is of lesser >>>>>> importance. >>>>> So, you have no answer. >>>> If something doesn't occur, the question "when does it occur" >>>> does not have an answer. >>> No, but I think the problem is elsewhere, slightly. Is there a formal >>> definition of what "transition" means? (Not in a nearby pocket "Dict. >>> of maths." for example) >>> >>> Seems to me that if you had the graph y = (1 if x<0; 2 if x>=0), and >>> and associated state transition diagram, then there would be a >>> "transition" from 1 to 2 "at" x=0. But such terminology does not >>> capture what the state is "at the point of" the transition, which may >>> be why it isn't used much. But if a vase has balls in it for values (-1 >>> <= t < 0), I can't see anything actually wrong with saying there is a >>> transition from empty to non-empty "at" t=-1 and a transition from >>> non-empty to empty "at" t=0. You need to be careful not to deduce >>> anything about the _state_ at the two transition values. >>> >>> After all, consider the graph of y = -x. This is positive for x<0, and >>> negative for x>0. It's undefined for x=0, but is there not a transition >>> from positive to negative at x=0? >> That all makes very good sense, Brian. I can't see that there isn't a >> point of transition, especially when that point is very explicitly >> defined to be noon, and that it is at least then, and not until then, >> that the vase goes from being non-empty to empty. > > Yes, but using this (slightly loose?) "transition" terminology, you > have to be careful that this tells you _nothing_ about the state _at_ > noon. Seems to me that all of the following graphs have a "transition" > from positive to negative at x=0, but very different things are true > _at_ 0 "_nothing_"? > > y=-1/x (at x=0 this is undefined) Or y=1/x. From one side it goes to oo, and from the other, -oo. This is resolved by application of the number circle concept, where oo=-oo. Then, while the value is "undefined" in the sense of being infinite, it's at least consistent with itself. > y = (1 if x<0; -1 if x>=0) : y(0) = -1 Clearly discontinuous - two different formulas > y = (1 if x<=0; -1 if x>0) : y(0) = 1 ditto > f = 1 if x<0; -1 if x>0; purple unicorn if x=0 : well, what can I say? > That's enough. None of those examples pertains to the vase, which is a very well defined increment of 9 per iteration. To put it in limit terms, the "solution" to the vase problem would be equivalent to saying lim(x->oo: 9x)=0. Ain't the case. > >> Is y=-x really "undefined" at x=0? I don't think so. It's 0, which >> equals -0. > > Sorry, that's a (surely obvious?) typo - I meant y=-1/x. No ok, perhaps > it isn't obvious. Okay, but oo can be viewed as equivalent to -oo, in some contexts. > > But another example: > y=-x : y(0) = 0, which is perfectly well defined, but neither positive > nor negative (au moins en anglais) > Yes, which fits perfectly well with 0's reciprocal, oo, or 100...000, which is also its own additive inverse. > > (I'll carry on with the blue sliver when I have time, which may be in a > day or so) > Brian Chandler > http://imaginatorium.org > Carry on! T |