An uncountable countable set
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From: Virgil on 29 Sep 2006 18:03 In article <451d6de7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Han de Bruijn wrote: > > Tony Orlow wrote: > > > >> Your axiom system is a farse. > > > > I'd rather think it is a farce. > > > > Han de Bruijn > > > > Ooops, yes, but it might as well be in Farsi. > > Tony Its all Greek to TO.
From: Virgil on 29 Sep 2006 18:05 In article <451d83c4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <8cabe$451ccd62$82a1e228$12622(a)news1.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >> Virgil wrote: > >> > >>> In article <9fd0$451b7e7b$82a1e228$8977(a)news1.tudelft.nl>, > >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>> > >>>> Virgil wrote: > >>>> > >>>>> In article <451a8f41(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>> For purposes of measure on the finite scale, infinitesimals can be > >>>>>> considered nilpotent. That's all. Do you disagree? > >>>>> I disagree that scale changes can convert between zero and non-zero. > >>>>> > >>>>> There are approximation methods is which products of small quantities > >>>>> are regarded as negligible in comparison to the quantities themselves, > >>>>> but they are always just approximations. > >>>> Crucial question: are those "approximation methods" part of mathematics? > >>>> I'll take Yes or No as a sufficient answer. > >>> They are a part of the applications of mathematics to things other than > >>> mathematics, so they are marginal. > >> That sounds like a smart answer, but I don't buy it. > >> Again: are those "approximation methods" part of mathematics? Yes or No. > >> > > > > They are attempts to bend the mathematics to accommodate the needs of > > the sciences, so one would have to say "Yes and No". > > Would you like syrup with your waffle? So what balls remain in the vase at noon, oh waffler extraordinary?
From: Virgil on 29 Sep 2006 18:07 In article <451d88ad(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.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. > > "[R]andy does not deny that the vase becomes empty". That sounds like it > occurs. > > > > > If I ask you what date you took a trip to Mars last year, > > would you have an answer? > > Does the vase become empty? > > > > >> And so it's not important. I see. > > > > I think you would consider a trip to Mars very important, > > but nevertheless you would not be able to tell me when this > > trip occurred. > > > > - Randy > > > > I hear it's kind of boring up there anyway. Buncha rocks and dust. Fit's TO's mathematics quite well!
From: Virgil on 29 Sep 2006 18:09 In article <451d8a3b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451d5d29$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <1159438112.240001.268540(a)m7g2000cwm.googlegroups.com>, > >>> mueckenh(a)rz.fh-augsburg.de wrote: > >>> > >>>> Dik T. Winter schrieb: > >>>> > >>>>> > The successor function *is* counting (+1). > >>>>> > >>>>> Wrong. > >>>> After a while you will have run out of the predefined successor, > >>>> unavoidably. > >>> If that were ever to happen, one would have discovered a largest > >>> possible number. But it does not ever happen, because for every set x > >>> there is a set UNION(x,{x}) which is its successor. > >>> > >> I believe Wolfgang is saying that, once you run out of the starting > >> known successive symbols of your language, your alphabet, you then have > >> to employ an actual number system, using those elements recursively. > >> Since alphabets are generally finite, you can never represent "infinite" > >> quantities, in terms of string length. > > > > One does not need to, as every finite natural is representable by a > > finite string. > > But not every finite real, such as pi. That's why WM says pi doesn't > really exist as a number. Not as a natural and not as a rational. But that is the case with "most" numbers. > Personally, I think it exists as a point on > the real line. What TO thinks is not evidence.
From: Virgil on 29 Sep 2006 18:12
In article <451d8b01(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451d5e15(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> 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. > >>>>> > >>>>> Han de Bruijn > >>>>> > >>>> 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. > >>> > >>> - Randy > >>> > >> I never said it did. When did I say that? I will offer this simple > >> logical argument. If the vase ever became empty, it would be because one > >> ball was removed, as per the gedanken, but 10 balls would have been > >> inserted immediately beforehand. The vase would therefore have had to > >> contain -9 balls, which I'm afraid is simply impossible. Don't you? It's > >> a ridiculous set-theoretical result. > > So is TO's conclusion that there ought to be infinitely many naturally > > numbered balls in the vase for which he can not find the number of any > > one of them. > > If you stuck aleph_0 consecutive balls in there, you have aleph_0/10+1 > through aleph_0 left in the vase. Which, not being numbered by allowable numbers, do not exist. in the original problem. So TO is cheating by bringing in wetbacks. > > > > > Let us consider a slightly modified experiment in which as each ball is > > removed from the vase, it is put into an initially empty urn. > > That was my idea a year ago. > > > > > Now at or after noon, ball 1 is in the urn (and not in the vase). > > Furthermore for every n-marked ball in the urn, ball n+1 is also in the > > urn. > > > > Thus, by induction, EVERY naturally numbered ball is in the urn (and > > not in the vase). > > > > So which balls are still in the vase and not in the urn, TO? > > What's amazing about that situation is that you are adding 10 balls to > the vase for every one you remove and put in the urn, but the urn ends > up with all the balls and vase becomes empty. When's your birthday? I'll > get you a label maker. That is what the rules of the game require as an end result. Don't blame the game for your own problems in understanding it. |