An uncountable countable set
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From: Randy Poe on 5 Oct 2006 10:21 Han de Bruijn wrote: > Randy Poe wrote: > > > Math has to be logical. It doesn't have to be physically > > realizable. > > Wrong. It doesn't have to be logical? > Math has to be an idealization which can be materialized again > into something that is physically realizable. According to you? Who put you in charge? > Quote: Physicists also > > realize that things can exist in mathematics that aren't even > > approximations of a physical realizable. That aren't physically > > sensible in other words. Correct. There is no requirement that a mathematical object have any physical meaning or applicability. > That's only true for non-disciplinary mathematics. What the heck is "non-disciplinary mathematics"? Do you mean "non-applied mathematics?" or "mathematics not modeling a physical system?" I'll agree with you that mathematics which doesn't model physics is not necessarily a model of a physical system. Is that all you're trying to say? Then we're in agreement: Some mathematical objects don't model physical systems. - Randy
From: Randy Poe on 5 Oct 2006 10:22 Han de Bruijn wrote: > Virgil wrote: > > > In article <161ca$4523bb80$82a1e228$8996(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>imaginatorium(a)despammed.com wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>The question is: how many balls are there in the vase at noon. > >>>>This question is meaningless, because noon is never reached. > >>> > >>>Really? When's lunch, then? > >> > >>Time is _suggested_, but not present, in the Balls in a Vase problem. > > > > Without any time to put balls into the vase, the vase is empty. > > In real time there are no singularities. With the Balls in a Vase there > is a singularity at noon. That's a property of the vase, not the clock. The clock ticks on independently of the imaginary process we're timing with it. - Randy
From: Randy Poe on 5 Oct 2006 10:24 Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > >>Mike Kelly wrote: > >> > >>>Appeals to dead authorities aside, you're choosing not to explain what > >>>is meant by "mathematics without the discipline". Why? > >> > >>Why not? > > > > It seems very silly to post seemingly silly messages to usenet and then > > refuse to explain what non-silly meaning they have behind them. People > > will think you utterly silly. > > No: What's the beef in explaining something that is self explanatory? If it's "self-explanatory" then wouldn't people who read it understand it? Do you think anyone reading this thread, beside you, understands what you mean by that phrase? - Randy
From: Dik T. Winter on 5 Oct 2006 10:25 In article <1160044514.105544.245260(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > > > (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? > > > > No, I never did. But you draw conclusions about it about the set N. > > Indeed, for each finite n, it is true. > > And N is nothing but the collection of all finite n. That does not make it true for N itself. > > You attack the existence of an infinite sequence, and so also the existence > > of an infinite set. So be it. But that is just a negation of the axiom of > > infinity. With that axioms such things do exist. > > The axiom of infinity does only state n+1 exists if n is given. It is > realized by he numbers of my list. That is *not* what the axiom of infinity states. The axiom states that there exists a set that contains all the successors of its elements. > > > Instead of "to index position" we can also say "to cover up to position > > > n". Hence you assert that it is possible to cover 0.111... up to every > > > position but it is impossible to cover every position. > > > > Yes. > > That is obviously a false claim, because "up to every (including this)" > means "every". At this point a further discussion is really fruitless. Yes, because you still do not understand that there is no "specific position" such that "every position" is the same as "upto that specific position". > > > Cantor's argument was about reals. He strived for generality but did > > > not see that two symbols are not enough. > > > > You are seriously wrong. > > Read his first paper. He treats numbers and rational functions. Nothing > more. Read the two first paragraph of his second paper more thoroughly. Or are you of the opinion that when I write: "In paper A I did prove the theorem that there are sets that have larger cardinality than the natural numbers. In this paper I will give a simpler proof of that theorem." you do not know for what theorem I will give a simpler proof in the current paper, but that you need first to read paper A to be able to state that? And you assert that it is not likely that in the current paper the theorem for which a simpler proof is given is the theorem "that there are sets that have larger cardinality than the natural numbers", but something else, unstated in the current paper? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 5 Oct 2006 10:31
In article <1160045362.894290.321140(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1159978513.826507.125470(a)m73g2000cwd.googlegroups.com> muecke= > nh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > > You can read it above. There is the order preserving union of a s= > et of > > > > > k negative numbers and a set of omega natural numbers including z= > ero. > > > > > > > > But that has ordinal k + omega, > > > > > > that is what I said! > > > > No. > > I used "k + omega " as the ordinal number of {-k, -k+1, ..., 0, > 1,2,3,...}. Sorry, I misread. > > > > not -k + omega. A set of k negative numbers > > > > has ordinal k; not -k. > > > > > > But subtraction of a set of positive numbers from the set omega is > > > expressed, as I did, by -k + omega. > > > > Ah, apparently you are defining something new here. > > New for you probably. As omega + k is different from k + omega, we > should not write omea - k for -k + omega. Cantor did. And he wrote omega_(-k) for the other. > > > No? Who decides that? You see, I knew already that, according to > > > Cantor, subtraction is possible. If I express this as addition of > > > negative, what do you think did I meant? > > > > I did not know because there is no definition presented nor available. > > Addition is defined between ordinal numbers. Ordinal numbers are (by > > their very definition) larger than or equal to 0. But you want to > > define addition between ordinal numbers and non-ordinal numbers. > > Wrong again. k and omega are ordinal numbers. But '-k' is *not* an ordinal number. In the meantime I have read it. You may note that in his notation the solution for x + a = b is b - a (a < b). And the solution for a + x = b (if that exists, and if there is more than one solution, the smallest one) as b_(-a). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |