An uncountable countable set
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From: Virgil on 29 Sep 2006 14:46 In article <d5a7a$451cc970$82a1e228$9055(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <cddac$451b7c18$82a1e228$8598(a)news1.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > > I am merely countering your nonsense that math without science is > > nonsense. > > That's what I'm saying all the time. That HdB says it all the time does not make it true. > > According to history, and what can be deduced of prehistory, there was a > > good deal of math, at least arithmetic, around long before there was any > > science at all. > > No. Because that "good deal of math" must be considered as scientific. Unless HdB is conflating "science" with lore, he is wrong. Counting is a part of lore, which precedes science by millenia.
From: Virgil on 29 Sep 2006 14:51 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".
From: Virgil on 29 Sep 2006 14:53 In article <4f664$451ccf5c$82a1e228$13885(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > Mathematicians live in the real world, but their creations are no more > > tied to reality than those of poets or writers of fantasy. > > There is a enormous difference between x^n + y^n = z^n and aleph_0 , > if (how mathematics be tied to the real world) is considered. Not for n > 2.
From: stephen on 29 Sep 2006 14:53 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Tony Orlow <tony(a)lightlink.com> wrote: >>> stephen(a)nomail.com wrote: >>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> stephen(a)nomail.com wrote: >>>>>> Randy Poe <poespam-trap(a)yahoo.com> wrote: >>>>>> >>>>>> <snip> >>>>>> >>>>>>> What is the number of the ball which, when removed, >>>>>>> makes the vase empty? >>>>>>> I know the kind of nonsense you will spout in answer to >>>>>>> those questions, but the answers within our axiom system >>>>>>> are: (1) there is no t<noon which is the moment just >>>>>>> before noon. For any t<noon, there is t < t' < noon. >>>>>>> (2) There is no such ball. >>>>>>> Here are the Tony gobbledgook answers: >>>>>>> (1) noon - 1/oo >>>>>>> (2) Ball number omega >>>>>>> In TO-matics, one can confidently give an answer like >>>>>>> number 2 despite the fact that one can also agree >>>>>>> that no ball numbered omega is ever put into the >>>>>>> vase. >>>>>> In TO-matics, it is also possible to end up with >>>>>> an empty vase by simply adding balls. According to TO-matics >>>>>> >>>>>> ..1111111111 = 1 + 1 + 1 + 1 + ... >>>>>> >>>>>> and >>>>>> ..1111111111 + 1 = 0 >>>>>> >>>>>> So if you just keep on adding balls one at a time, >>>>>> at some point, the number of balls becomes zero. >>>>>> You have to add just the right number of balls. It is not >>>>>> clear what that number is, but it is clear that it >>>>>> exists in TO-matics. >>>>>> >>>>>>> But in mathematics and logic, we don't get to >>>>>>> keep a set of self-contradictory assumptions around, >>>>>>> only using the ones we want as needed. >>>>>>> - Randy >>>>>> Where's the fun in that? :) >>>>>> >>>>>> Stephen >>>>> You drew that from my suggestion of the number circle, and that ...11111 >>>>> could be considered equal to -1. Since then, I looked it up. I'm not the >>>>> first to think that. It's one of two perspectives on the number line. >>>>> It's either really straight, or circular with infinite radius, making it >>>>> infinitesimally straight. The latter describes the finite universe, and >>>>> the former, the limit. But, you knew that, and are just trying to have fun. >>>>> Tony >>>> I am just pointing out that according to your mathematics >>>> that if you keep adding balls to the vase, you can end up >>>> with an empty vase. The fact that other people may have >>>> considered a number circle does not change the fact that the >>>> number circle implies that if you keep on adding balls, eventually >>>> you will have zero balls. >> >>> That's a bastardization of the concept. There are two ways to look at >>> the number circle, and you are combining them in mutually contradictory >>> ways. >> >> How is it a bastardization of the concept? You claim that >> 1+1+1+1+ ... = ..11111111 > I wouldn't put it that way. ...1111 can be interpreted as the largest > binary natural, if you claim all bit positions are finite. I do claim that all bit positions are finite, but there is no largest binary natural. That is something you have invented. So this "largest binary natural" of yours is not equal to 1+1+1+... for some number of 1's? In other words, it is not in the chain of successors? In what sense then is it a "natural number"? >> 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. I know of now computer that can hold ...111111 let alone intreprets it as -1. Remember, ...111111 is an unending string of 1's. > Those are two different interpretations of -1. They aren't really > compatible, as far as I can see, although Ross used to like to point out > that temperatures below 0 Kelvin were somehow hotter than any positive > temperature. There could be a connection. The moment you mention physics or Ross it is clear you are just rambling. >> >> Why does that not apply to balls in the vase? Each ball >> is a 1. If a add balls, I add 1's. Do I not eventually get 0? >> If it does not apply to balls, what does it apply to, >> and how do you determine when it applies? >> > They are two different interpretations of the string. Do you honestly > think that I think sum(x=1->oo: 1)=0? For god's sake, how long have I > been saying it's infinite, and that's why any infinite set of naturals > would have to include infinite naturals? Oy. Like I said, you're > combining concepts that are mutually contradictory, from two different > number systems. And, people complaint hat I try to use normal operations > normally on infinite numbers.... You are the one who is combining things. In "my" system ...111111 does not exist. Yet it does in your system, and it apparently is a natural number, and it apparently equals 1+1+1+.... (does it not need to be a successor of a successor of a successor and so on of 1), and it apparently sometimes equals -1. Can you clearly and concisely state the rules that govern ...11111111? If not, please never mention it again. >>>> So why is it okay to end up with zero balls, when you never remove >>>> any at all, but it is not okay to end up with zero balls when >>>> each ball is clearly removed at a definite time? >> >>> Because the model of the number circle where all strings are positive is >>> incompatible with the number circle where any string with a >>> left-unending string of 1's (in binary) is negative. Duh. >> >> That seems to be a non-answer. The most I can glean from >> that is that the number circle is not relevant to the balls >> in the vase problem. Is the number circle relevant to anything? And >> how does one determine when it is relevant? If it is not >> relevant to anything, why did you bring it up in >> the first place? > There are an number of topics going on here besides the crazy vase. It's > relevant to number systems, and came up initially with my suggestion > that ...1111 represents the largest finite if all bit positions are > finite. I only mentioned the alternative interpretation of ...1111 being > -1 as a sidebar. It's an *alternative* interpretation of the *string*. Okay. So the number circle is not relevant to this problem, and ...11111 is not equal to -1 in this context. I have no problem with that. However elsewhere you seemed to be arguing that there
From: Randy Poe on 29 Sep 2006 14:56
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. If I ask you what date you took a trip to Mars last year, would you have an answer? > 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 |