An uncountable countable set
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From: Han de Bruijn on 2 Oct 2006 04:36 Virgil wrote: > In article <451d66c0(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >>stephen(a)nomail.com wrote: >> >>>Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> 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. >>> >>>Yes it is a nonsense question, in the sense >>>that it is non-physical. You cannot actually perform >>>the "experiment". Just as choosing a number uniformly >>>from the set of all naturals is a non-physical nonsense >>>question. You cannot perform that experiment either. >> >>Yes, they both sound equally invalid, and it all goes back to omega, but >>Han has a point about the density of the set in the naturals throughout >>its range, and the overall statistical probability of selecting one of >>that subset from the naturals, even if having probabilities of 1/omega >>for each natural presents problems. > > Do statistical probabilities have to satisfy the condition that their > sum over all indivisible outcomes must equal 1? Precisely! Inasmuch as the sum over all indivisible outcomes must equal 1 in the limit of the Riemann sum which represents the integral(0,1)dx. Han de Bruijn
From: Han de Bruijn on 2 Oct 2006 04:54 Virgil wrote: > In article <1159727459.165196.109230(a)b28g2000cwb.googlegroups.com>, > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>Randy Poe wrote: > >>>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. > > Who said anything about this being "nature"? > "Nature" does not have an endless supply of balls in the first place, Definitely. And "Nature" does also not have an endless supply of dx's in integral(0,1)dx = 1 . Yet this integral is of significance to physics. >>>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 ... > > Leave it to a physicalist to insist that a nonphysical problem is > physics. According to Virgil's criteria, most of theoretical physics would be unphysical. Han de Bruijn
From: Han de Bruijn on 2 Oct 2006 05:14 Tony Orlow wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >> Randy Poe wrote: >> >>> Noon exists. >> >> Sure. By dogma. Randy Pope is infallible. > > See? Your position now is that noon doesn't exist. Yes. That has been my position all the time. Consider the ideal gas law: p.V = constant ; p = pressure, V = volume. Imagine a finite container with gas and start compressing. Then V is the independent and p is the dependent variable. The more you compress (V <) the higher the pressure (p >). Can the volume ever become zero (V = 0) ? No, it can't. So a zero volume doesn't exist in this problem. Mind that V is the _independent_ variable, though! >>> 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. > > What's that, a type of mud? It's English-Dutch for "het is zo klaar als een klontje" => "it is as plain as the nose on your face". See? (I can be more espressive in my mother's tongue) >> Nature does not jump, Leibnitz said. > > Leibniz lived in reality. Is that a necessary requirement? Yes. Typo. >>> 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 ... > > The time was not before that moment, but it was not after it either. > Therefore we cannot make any judgments as to when it happened. That's > transfinitology for you. Uhm? Dunno what you mean. Han de Bruijn
From: Han de Bruijn on 2 Oct 2006 06:05 Virgil wrote: > In article <65e14$4520c63d$82a1e228$23833(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Virgil wrote: >> >>>In article <e4ca4$451cd0dd$82a1e228$14108(a)news1.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>>Virgil wrote: >>>> >>>>>In article <1159437062.473100.294820(a)k70g2000cwa.googlegroups.com>, >>>>>mueckenh(a)rz.fh-augsburg.de wrote: >>>>> >>>>>>Virgil schrieb: >>>>>> >>>>>>>Several sets may all have the common property of being pairwise >>>>>>>bijectable, but if any of their members are distinguishable from those >>>>>>>of another set then the sets are equally distinguishable. >>>>>> >>>>>>Each one of the sets expresses, represents, and *is* the same >>>>>>(cardinal) number. >>>>> >>>>>Then one apple and one orange are the same because they have the same >>>>>cardinality. >>>> >>>>In _that_ respect, with respect to counting: definitely, yes! >>> >>>But not necessarily in any other respect whatsoever, so that to say an >>>apple is an orange or an orange is an apple, as some have been saying, >>>is foolishly wrong. >> >>Why? Give me one piece of fruit. I don't care whether it is an orange or >>an apple .. > > A bunch of grapes is one piece. Yes. If you want an equivalence class, that's allright. Han de Bruijn
From: Han.deBruijn on 2 Oct 2006 06:29
Virgil wrote: > The problem as I recall it was this: > > Given an infinite set of balls numbered with the infinite set of > naturals and an "infinitely large" initially empty vase, and a positive > time interval in seconds, t, and a small positive time interval in > seconds, epsilon ( much smaller than t/2). > (1) At time t before noon balls 1 through 10 are put into the vase and > at time t - epsilon before noon ball 1 is removed. > (2) At time t/2 before noon balls 11 through 20 are put into the vase > and at time (t - epsilon)/2 before noon ball 2 is removed. > ... > (n) At time t/2^(n-1) before noon balls 10*(n-1)+1 through 10*n are put > in the vase and at time (t-epsilon)/2^(n-1) before noon, ball n is > removed. > ... > > The question is what will be the contents of the vase at or after noon. There is no noon in this problem. Or: the problem is _undefined_ at noon. > Of course, there can be no physical analog of this thought experiment, Not a single thought experiment has a "physical analog" as you recall it, but there are _many_ highly abstract thought experiments which can nevertheless be attached to a physically meaningful arrangement. In this problem, things remains meaningful through all times < noon. > but in mathematics the question has a clear answer. You think so. But it's just an illusion. Han de Bruijn |