An uncountable countable set
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From: mueckenh on 19 Sep 2006 07:46 Mike Kelly schrieb: > > All naturals do not exist. What is "all"? > > Huh? So some naturals don't exist? What does that mean? How can > something that doesn't exist be a natural number? Not all novels do exist. Is that meaning clear to you? > > You are being dishonest by misrepresenting what people say to support > your position. > Isn't that sentence applicable to your above question too? Regards, WM
From: mueckenh on 19 Sep 2006 07:53 Mike Kelly schrieb: > > What bearing does empirical evidence have on what is mathematically > meaningful? If mathematics states that III + II = IIIII, then it is true (at least the result), and if it states otherwise, then it is false. That is very easy and needs no axioms. Regards, WM
From: mueckenh on 19 Sep 2006 08:08 Mike Kelly schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Mike Kelly schrieb: > > > > > > Any set that can be established is a finite set. > > > > > > Why? > > > > Look: If aleph_0 were a number larger than any natural number, then for > > any natural number n we had n < aleph_0. "For all" means: even in the > > limit. > > OK so far. Every cardinal number which is a natural number is less than > aleph_0. > > > So lim [n-->oo] n/aleph_0 < 1 > > Division is not defined for infinite cardinal numbers. Is that your only escape? If you dare to say that aleph_0 > n for any n e N, then we can conclude the above inequality. But remedy is easy. Take lim [n-->oo] aleph_0 / n > 1 and reverse the following fractions analogously. > > > If aleph_0 counted the numbers, for instance the even naturals, then we > > had for all of them > > > > lim [n-->oo] |{2,4,6,...,2n}| = aleph_0. > > > > This would yield lim [n-->oo] (2n/|{2,4,6,...,2n}|) < 1 > > > > But we have lim [n-->oo] (2n/|{2,4,6,...,2n}|) = 2 > > > > Therefore aleph_0 does not exist as a number which could be compared > > with other numbers. > > > > > > > No. Just this is the point! The series 1 + 1/2 + 1/4 + ... is 2 (or at > > > > least as close to 2 as we like), not by definition and not by any > > > > axiom, but by rational thought. > > > > > > Prove that to be the case without using any definition of what a series > > > is and without any axioms. > > > > Archimedes did so when exhausting the area of the parabola. In decimal > > notation 2 + 2 = 4, and in any system we have II and II = IIII. > > In airthmetic modulo 3, 2+2 = 1. If you say "in arithemtic mod 3", then you imply that you subtract 3 from the true result as often as possible. It does not invalidate II + II = IIII, if you subsequently tale off III. > > >For self-evident truths you don't need axioms. Only if you want to > > establish uncertain things like "There exist a set which contains O and > > with a also {a}" then axioms may be required. > > > > Don't misunderstand me: I do not oppose the principle of induction but > > the phrase "there exists" which suggests the existence of the completed > > set. > > Why do you object to this? Because of he proof above. Regards, WM
From: Mike Kelly on 19 Sep 2006 09:44 mueckenh(a)rz.fh-augsburg.de wrote: > Mike Kelly schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Mike Kelly schrieb: > > > > > > > It is meaningful to say that a natural drawn uniformly at random from a > > > > set of consecutive naturals 1 thru 3n has a 1/3 probabaility of being > > > > divisible by 3. Nobody disputes this. But talking about the probability > > > > of "a natural" being divisible by 3 implies a uniform distribution over > > > > the naturals. Such a thing does not exist. > > > > > > Talking about sinx / x for x --> 0 does not imply the existence of sin0 > > > / 0. Neither does the result 1/3 imply the distribution for a realy > > > infinite set f naturals. There is no real (actual, finished) infinity, > > > neither in physics > > > > Ok. > > > > >nor in mathematics. > > > > Why? > > Because all we are and all we think with (brain, neurons, currents, > loads, and ideas, letters, words, pictures, i.e., hardware and > software) and all we think and all we think we are thinking: all that > is physics. So it is impossible to think of something that does not physically exist? -- mike.
From: Mike Kelly on 19 Sep 2006 10:18
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: > >>>>> > >>>>>>Mike Kelly wrote: > >>>>>> > >>>>>>>Han de Bruijn wrote: > >>>>>>> > >>>>>>>>Mike Kelly wrote in response to Tony Orlow: > >>>>>>>> > >>>>>>>>>*sigh*. Probabilities are *standard* real numbers between 0 and 1. > >>>>>>>> > >>>>>>>>Yes. And infinitesimals are *standard* real numbers in engineering. > >>>>>>> > >>>>>>>Engineering is not mathematics. It uses mathematical results. > >>>>>> > >>>>>>Sure. And moslems are not praying. Only roman catholics do. > >>>>> > >>>>>Bizarre and borderline offensive analogy. Engineering isn't > >>>>>mathematics, anymore than accounting is mathematics. They both *use* > >>>>>mathematics. > >>>> > >>>>Wrong. They both *create* mathematics as well. I know, because I've been > >>>>active in both engineering and adminstration. > >>> > >>>But not in mathematics, evidently. So how do you tell the application > >>>of mathematics from mathematics itself, then? > >> > >>Good question! I find it impossible to distinguish them, while it seems > >>that you can do it. Unambiguously ? Please tell me how, because I don't > >>understand. > > > > Mathematics is the derivation of theorems from axioms. The application > > of mathematics is using the theorems of some mathematical theory to > > solve problems that the theory can model. > > Then I am right and your so-called "applications" are a form of active > mathematics. What is "active mathematics"? >Mathematicians simply can not and do not come up with all > the theorems most modern applications are in the need of. So we develop > them ourselves. Whether this situation is desirable is another question. I'm not really seeing the point in this line of discussion. In what way does any of this dispute that probabilities are not infinitesimal? > >>>>>>>>That's why infinitesimal probabilities will become feasible as soon > >>>>>>>>as mathematics becomes a science which is compliant with engineering. > >>>>>>> > >>>>>>>Mathematics is not a science. What exactly would it mean for it to > >>>>>>>"become a science compliant with engineering"? > >>>>>> > >>>>>>Ah! Mathematics is not a science. So mathematics is not serious at all! > >>>>>>Why didn't you tell me this before? > >>>>> > >>>>>How do you get from "mathematics is not a science" to "mathematics is > >>>>>not serious"? Time to brush up on your English, Han. > >>>> > >>>>How can mathematics be serious, if it is not scientific? > >>> > >>>Definately a translation issue. Can I expect a lecture from you on the > >>>proper meaning of the word "science", now? > >> > >>Not now. But _you_ started saying that "Mathematics is not a science". > >>How can you be so sure, if you need an explanation from me? > > > > I don't need an explanation from you. You really can't read English, > > can you? > > You can't read Dutch, can you? I'm not attempting to argue in Dutch, am I? -- mike. |