An uncountable countable set
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From: Tony Orlow on 29 Sep 2006 21:59 Virgil wrote: > In article <451da475(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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. >>>>> >>>>> Stephen >>>> 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. >>>> >>>> Tony >>> Do statistical probabilities have to satisfy the condition that their >>> sum over all indivisible outcomes must equal 1? >> Yes, and that requires, for a uniform distribution, that we have a >> precise count. If aleph_0 is some kind of precise number, then we can >> say that a uniform probability distribution over a set of this size >> yields a probability of 1/aleph_0, and if aleph_0 is infinite, this >> individual probability is infinitesimal. > > Except that this does not happen within the standard reals of > probability theory.\ God! No Duh! Like, as IF! In your dreams... ;) >> Then you can add them all up and get the unit of universal inevitability. :)
From: Tony Orlow on 29 Sep 2006 22:09 Virgil wrote: > In article <451da3b6(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <451d6602(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>> >>>> I wouldn't put it that way. ...1111 can be interpreted as the largest >>>> binary natural, if you claim all bit positions are finite. >>> Calling all bit positions finite does not require that there only be a >>> finitely many bit positions, and the binary string representation of >>> every finite natural n requires <= ln(n+1)/ln(2) bit positions. >>> >> If all bit positions are finite, and the string up to any finite bit >> position can only have a finite value, then there is no position in the >> string where it achieves anything but a finite value. > > Precisely! Does the string comprise anything except for these bits? It seems to me that it doesn't happen within the string,. so the string does not represent that event or situation. >> Are you saying that aleph_0 naturals only require ln(aleph_0+1)/ln(2) >> bit positions? > > Not at all. I am talking about indvidual natural numbers as members of > N, ,not N itself, which is not a member of N. > > And also for every set of contiguous naturals starting at 0 EXCEPT for N. Why EXCEPT for N? >>>>> and that >>>>> ..11111111 + 1 = 0 >>>> ...11111 can be interpreted indeed as -1, as is done every millions of >>>> times per microsecond all over the world in computers. >>> Which of the worlds computers can work with an infinitely long string of >>> binary digits? >> The fact works for an arbitrary number of bits, including in the 2-adics. > > Irrelevant. That does not tell me anything about which , if any, actual > computers deal with infinitely long strings of binary digits. Like, none, man, unless you droppa lotta 'cid, dude. Isn't this math? We're discussing an infinite limit of a situation that each of us have caused about a trillion times in this conversation alone. >> Besides, any Turing machine can process an infinite string given >> infinite time. > > Go watch one, and don't come back till its finished. Alright....
From: Tony Orlow on 29 Sep 2006 22:12 Virgil wrote: > 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? For n balls inserted, balls n/10+1 through n remain at the end of any iteration n. You specify the number of iterations, I'll give you the sum. What was it? Aleph_0?
From: Tony Orlow on 29 Sep 2006 22:21 Virgil wrote: > In article <451d8c58(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <451d6037(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> 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. And so it's not important. I see. >>> When have all the balls, whether removed or not, all been inserted? >> Not until noon. >> >>> That is the time at which all have also been removed, as no ball can be >>> removed before its insertion nor after all have been inserted. >> Repeat that to yourself 10 times, and you will see God. > > Whose notion of god? TO's? Thank you, no. I prefer a god who works > within logic. Uhhhh.... That was YOUR statement. But okay...... >>> Except for the first 10 balls, each insertion follow a removal and with >>> no exceptions each removal follows an insertion. >> Which is why you have to have -9 balls at some point, so you can add 10, >> remove 1, and have an empty vase. Can you have -9 balls in your vase? Is >> it like a ball bank account that can be overdrawn? Are you sure it's not >> Cantor's ashes in there? > > TO deludes himself with the notion that what does not contain an ending > must contain an ending. For him all intervals are closed intervals. You certainly seem to me to have very well defined bounds, but I'm not sure you should project that on all my opinions about reality. It's more of a personal observation.
From: Tony Orlow on 29 Sep 2006 22:26
Virgil wrote: > In article <451d8f8d(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Randy Poe wrote: >>> Tony Orlow 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? >>> Virgil and I differ on terminology here. As I have already said, >>> you are trying to pin down an identifiable pair of contiguous >>> moments where the vase is non-empty in one, and empty >>> in the next. As I have already said, a verb like "emptying" >>> conveys to me the existence of a PAIR of moments with >>> that property, of an identifiable "change moment". I would >>> not use the word "become" for the same reason. >>> >>> So I will continue to say what I have said. The vase is empty >>> at noon, because before noon every ball put in was taken >>> out. >>> >>> There is no moment when the vase "becomes empty". The >>> first time when the vase IS empty is noon. >>> >>> - Randy >>> >> Sorry, Randy, that's just daft. It's not, then it is, but it didn't >> become, because that would mean you'd have to think about that moment of >> becoming, and face the fact that you would need to have -9 balls two >> iterations beforehand to get an empty vase. Can you say "avoidance"? > > > > What Randy is avoiding is TO's fallacious insistence that there must be > a last ball removed if one is ever to achieve a state where the balls > are all removed. When the gedanken specifically states that only one ball is removed at a time, what is fallacious about that statement? You are the one chanign the game in the middle. > > In a physical world that might be the case, but in an ideological one it > need not be. It is stated as a condition. Insert 10, remove 1, repeat. > > When one plays out a thought experiment. one must play it out by the > rules given in that experiment. Find your nearest mirror, face it, and repeat. > > To tries to change, or break, those rules, which is a form of cheating. Oh, by rearranging a sequence and violating a specified order? No, that was your trick. Tony |