An uncountable countable set
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From: mueckenh on 5 Oct 2006 06:49 Dik T. Winter schrieb: > In article <1159978513.826507.125470(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > You can read it above. There is the order preserving union of a set of > > > > k negative numbers and a set of omega natural numbers including zero. > > > > > > 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,...}. If you can't understand that, try to increase your knowldge of set theory. > > > > 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. > > > > > Now try to > > > > > apply that definition with -k, where k is positive. What is a set > > > > > with ordinal "-k"? Do you know how ordinal addition works? > > > > > > > > You can find it in Cantor's collected works, e.g., p. 201: "Die > > > > Subtraktion kann nach zwei Seiten hin betrachtet werden." And you can > > > > read above how it works in this special case. The first k elements are > > > > cut off. > > > > > > And that, again, is quite different from what you did write. But I will > > > read and see how Cantor does define omega - 1. A left-handed subtraction > > > is indeed possible, but that is not the same as a left-handed addition by > > > a negative number. > > > > 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. > Pray > go ahead, and supply your definitions. (I still have not found the time > to see how Cantor defined omega - 1.) > Here it is, but some greek symbols may be misprinted. Die Subtraktion kann nach zwei Seiten hin betrachtet werden. Sind ï?¡ï? und ï?¢ irgend zwei ganze Zahlen, ï?¡ï? < ï?¢, so überzeugt man sich leicht, daÃ? die Gleichung ï?¡ï? + ï?¸ = ï?¢ immer eine und nur eine Auflösung nach ï?¸ zuläÃ?t, wo, wenn ï?¡ï? und ï?¢ Zahlen aus (II) sind, ï?¸ eine Zahl aus (I) oder (II) sein wird. Diese Zahl ï?¸ werde gleich ï?¢ï? - ï?¡ gesetzt. 202 Betrachtet man hingegen die folgende Gleichung: ï?¸ï? + ï?¡ = ï?¢ so zeigt sich, daÃ? dieselbe oft nach ï?¸ gar nicht lösbar ist, z. B. tritt dieser Fall bei folgender Gleichung ein: ï?¸ï? + ï?· = ï?· + 1. Regards, WM
From: Han de Bruijn on 5 Oct 2006 06:58 Mike Kelly wrote: > Han de Bruijn wrote: > >>Mike Kelly wrote: >> >>>Han de Bruijn wrote: >>> >>>>Quote [ Randy Poe ] : 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. >>>> >>>>That's only true for non-disciplinary mathematics. >>> >>>What is non-disciplinary mathematics? Is it not mathematics? >> >>It is mathematics without the discipline. [ ... First guess snipped ... ] > Second guess was "mathematics that is not liked by Han". Not liked by Carl Friedrich Gauss as well. I'm in fairly good company. Han de Bruijn
From: Mike Kelly on 5 Oct 2006 07:08 Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > >>Mike Kelly wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>Quote [ Randy Poe ] : 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. > >>>> > >>>>That's only true for non-disciplinary mathematics. > >>> > >>>What is non-disciplinary mathematics? Is it not mathematics? > >> > >>It is mathematics without the discipline. > > [ ... First guess snipped ... ] > > > Second guess was "mathematics that is not liked by Han". > > Not liked by Carl Friedrich Gauss as well. I'm in fairly good company. Appeals to dead authorities aside, you're choosing not to explain what is meant by "mathematics without the discipline". Why? -- mike.
From: Han de Bruijn on 5 Oct 2006 07:17 Mike Kelly wrote: > Han de Bruijn wrote: > >>Mike Kelly wrote: >> >>>Han de Bruijn wrote: >>> >>>>Mike Kelly wrote: >>>> >>>>>Han de Bruijn wrote: >>>>> >>>>>>Quote [ Randy Poe ] : 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. >>>>>> >>>>>>That's only true for non-disciplinary mathematics. >>>>> >>>>>What is non-disciplinary mathematics? Is it not mathematics? >>>> >>>>It is mathematics without the discipline. >> >>[ ... First guess snipped ... ] >> >>>Second guess was "mathematics that is not liked by Han". >> >>Not liked by Carl Friedrich Gauss as well. I'm in fairly good company. > > Appeals to dead authorities aside, you're choosing not to explain what > is meant by "mathematics without the discipline". Why? Why not? Han de Bruijn
From: Mike Kelly on 5 Oct 2006 07:36
Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > >>Mike Kelly wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>Mike Kelly wrote: > >>>> > >>>>>Han de Bruijn wrote: > >>>>> > >>>>>>Quote [ Randy Poe ] : 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. > >>>>>> > >>>>>>That's only true for non-disciplinary mathematics. > >>>>> > >>>>>What is non-disciplinary mathematics? Is it not mathematics? > >>>> > >>>>It is mathematics without the discipline. > >> > >>[ ... First guess snipped ... ] > >> > >>>Second guess was "mathematics that is not liked by Han". > >> > >>Not liked by Carl Friedrich Gauss as well. I'm in fairly good company. > > > > 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. -- mike. |