An uncountable countable set
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From: Virgil on 22 Sep 2006 13:42 In article <b0162$451388ff$82a1e228$17723(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <4fd34$45123f60$82a1e228$7325(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>stephen(a)nomail.com wrote: > >> > >>>Assuming that Tony's definition of infinitesimal has anything > >>>to do with any standard definition of infinitesimal. :) > >> > >>What is "any standard definition of infinitesimal" ? > > > > Any of these in "non-standard analysis" as initiated by Abraham > > Robinson, or "hyperreal numbers". > > How can "non-standard" be "standard"? > > Han de Bruijn Standards change. What was originally known as non-standard analysis has become a part of the standard with time. After all, Robinson published "Non-Standard Analysis" in 1966, so it isn't exactly new.
From: Virgil on 22 Sep 2006 13:51 In article <6eae4$45138993$82a1e228$17723(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <73452$45123e3f$82a1e228$7325(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>Virgil wrote: > >> > >>>In article <3a6c6$4510f00a$82a1e228$27505(a)news2.tudelft.nl>, > >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>> > >>>>David R Tribble wrote: > >>>> > >>>>>mueckenh wrote: > >>>>> > >>>>>>>Nothing has changed. There is no complete set of natural numbers. Any > >>>>>>>set that can be established is a finite set. Hence, the probability to > >>>>>>>select a number divisible by 3 is 1/3 or very very close to 1/3. > >>>>> > >>>>>Virgil wrote: > >>>>> > >>>>>>That presumes that the allegedly finite set of naturals that can be > >>>>>>constructed is nearly uniform with respect to divisibility by 3 at > >>>>>>least, and probably by other numbers as well. What is the justification > >>>>>>for this assumption? > >>>> > >>>>Wolfgang says litteraly: "_or_ very very close to 1/3". > >>> > >>>Which requires "_nearly uniform_ with respect to divisibility by 3". > >> > >>Sorry, Virgil. I don't get it. > > > > As usual. > > Other people in this group would have expressed "I don't get it" as > "You are an idiot". Do you get it now? > > Han de Bruijn If the probability of choosing a particular sort of member at random from a finite set is 1/3, or very very close to 1/3, then the number of that particular sort had better be 1/3 ,or very very close to 1/3, of the size of the set. But why should it be? Is one to assume that every natural up to some maximum is to be a member of that supposedly finite set of actually constructed naturals. Seems quite UNnatural to me. Do you get it now?
From: Virgil on 22 Sep 2006 13:57 In article <bf6b$45138cf1$82a1e228$22274(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > imaginatorium(a)despammed.com wrote: > > > stephen(a)nomail.com wrote: > > > >>Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: > >> > >>>In article <eeu7fn$ti$2(a)news.msu.edu> stephen(a)nomail.com writes: > > > >>> > I do not think Tony's infinitesimals are nil-potent. > >> > >>>But see what Tony did write! > >> > >>Yes. He has apparently changed his mind, or the properties > >>of his infinitesimals depend on his current argument. > > > > Isn't that the crux? Crank-infinitesimals are quantities that are zero > > when being nonzero would cause a contradiction, but nonzero when being > > zero would. > > Mainstream mathematics does not understand what infinitesimals are. > That is because the recognition of infinitesimals would be suicidal > to mainstream mathematics and its illusionary "rigour". There are perfectly rigourous mainstream infinitesimals, a la Abraham Robinson and others like him, and they cause no difficulties for mainstream mathematics at all. It is the crank-infinitesimal creators, whose systems are self-destructive, and their followers, who do all the complaining about being discriminated against when the are merely being treated like everyone else by being asked to validate their claims.
From: Dik T. Winter on 22 Sep 2006 21:02 In article <bf6b$45138cf1$82a1e228$22274(a)news1.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: .... > > Isn't that the crux? Crank-infinitesimals are quantities that are zero > > when being nonzero would cause a contradiction, but nonzero when being > > zero would. > > Mainstream mathematics does not understand what infinitesimals are. > That is because the recognition of infinitesimals would be suicidal > to mainstream mathematics and its illusionary "rigour". You do not understand either. (Wasn't it Leibniz who has said that he used them but did not know how to define them, or actually what they were, and wasn't it Robinson who actually gave a rigorous definition?) BTW, quite a long time ago I posted the message that numerical differentiation was more difficult than numerical integration. You vehemently contested that, and you did show a web page doing numerical differentiation, using non-mainstream mathematics. I offered a proof that both your and mainstream mathematics methods to determine the differential at some point suffered the same, if the grid became to small, the precision would deteriorate. Moreover, I did show that mainstream methods would give equivalent results with fewer sampling points than your method. You promised to look into that. Have you looked into it? I ask because it took me quite some time (using mainstream methods) to set up the comparison, and I have the impression that that time was just wasted as you do not want to hear otherwise than your opinion. But whatever, that experience did not encourage me to look at your later mathematical methods at all. If you do not want to hear an answer and just ignore it when it does not fit in your mind, much effort in responding to you is just wasted. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Han.deBruijn on 23 Sep 2006 14:38
MoeBlee wrote: > Han de Bruijn wrote: > > to mainstream mathematics and its illusionary "rigour". > > The rigor is in a recursive axiomatization with recursive rules of > inference. You haven't shown that this is an illusion. That rigor turns out to be incompatible with useful infinitesimals. Han de Bruijn |