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From: Ross A. Finlayson on 18 Apr 2005 05:24 Hello, That's a good point, about the divisibility of the continuum vis-a-vis infinitesimals as fixed iota-values. Basically that is from saying, "if a is the smallest positive real, then as the reals are a field a/2 is smaller, positive, and real, thus a was not the smallest positive real, contradiction, Q.E.D." A resolution of this notion derives from saying that a, or iota, while a real number, is indefinite in terms of the field of real numbers (eg each real and each pair of real's sum/difference and product/quotient) but as there are everywhere and only real numbers between two real numbers, the OTHER consequence of continuity, that there must be those iota-values. The reals as the continuum have multiple roles to play. That's where 2 is a real number. That gets into regular talk about infinitesimals, but with some specifics about some "least" positive real. After some infinite amount of iterations from a given "definite" real", eg 0, there is another given positive real, eg "1" or "2", but "2" only at some point _after_ 1, and correspondingly after twice as many iterations. Perhaps I am not very well understanding your argument, but you seem to be saying that the continuous line and the point are incompatible. I'm trying to promote the notion that they are not, and that indeed analytical results exists between the domain of the set of natural integers and the range of the set of the unit interval of real numbers, or N/U EF the natural/unit equivalency function. Here's another way to consider the continuum, it has to do with a concept from signal processing called the Nyquist frequency. The frequency of the reals, on the continuous real number line, is in a sense the Nyquist frequency of that of the rationals, perhaps. I think it may be, and I don't know, that when Martin was referring to successors of real numbers that he meant an ordering different than the usual total ordering of the set of positive reals, which as a set of integral iota-multiples, is called by myself its natural well-ordering. If not, I'm surprised, because that's not what people are taught, and he would basically had to have figured that out for himself, but not very. Then again, I claim Dedekind/Cauchy is inadequate to represent the reals, that reals are computable (although that's tentative), that reals are defined as a coincident or concurrent, confluent, synfluxious, contiguous sequence and field, consequent of the number system in the null axiom theory. Ross
From: Eckard Blumschein on 18 Apr 2005 06:48 On 4/16/2005 11:27 PM, Will Twentyman wrote: >> >>>but >>>his fundamental concepts were sound >> >> Quantifying a quality is not a sound concept. > > We do it all the time. Both ground beef and steak have the quality > "tasty", yet I can quantify which is *more* tasty. One way is to > establish a price/kg(or pound). Quantifying the qualities "useful", > "desireable", etc are the basis of economics. Well, my expression was not properly choosen. I shoud better have chosen "such a quantity" instead. The quality "infinite" is an either-or-quality, something that cannot be quantified like pregnant, dead, entirely, countable etc. Different levels of size of infinity are nonsense. The notion of infinite numbers is self-contradicting. Cantor felt entitled to create a nonsense-mathematics. This is nothing uncommon at all. However, after more than a hundred years it is legitimate to ask for at least any tiny benefit. So far I only got aware of trouble. >> I see most of the paradoxes just superficially remedied. > > Please name one that is not. I'll admit that some are not remedied in a > way that is entirely satisfying, but if the paradox is eliminated, where > is the superficiality? E. g., AC does not provide access to a well-ordered list of the rationals. > >>>I'll give one example where you are simply wrong: "Cýs infinite alephs >>>only distinguish between countable and uncountable sets." >> >> I am aware that they claim to manage much more. I referred to what they >> really are able to perform. > > They distinguish between more than two classes of sets. Different classes were just introduced in order to have a justification to introduce them? >> Given it would be make sense to subdivide the class of uncountable sets. >> Do you expect these subdivition nearly equally fundamental as the >> division into countable and uncountable numbers? > > Yes. What is not fundamental about taking the notion of dividing > infinite sets into the classes "countable" and "uncountable", and then > repeating the process on "uncountable" to give aleph_1 and aleph_2, then > aleph_3, then aleph_4...? Already the words countable and uncountable mutually exclude each other. I do not see any sense in this formal repetition that contradicts to the notion of oo. 130 years of missing benefits seems to confirm my doubt. The fact that the basic process can be > repeated is nearly as fundamental as the initial split between countable > and uncountable. Well. It must not be considered a brilliant idea. It is rather stupid. Nonetheless, it lacks any justification. E.
From: Dik T. Winter on 18 Apr 2005 08:01 In article <42639096.600(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: .... > > Please name one that is not. I'll admit that some are not remedied in a > > way that is entirely satisfying, but if the paradox is eliminated, where > > is the superficiality? > > E. g., AC does not provide access to a well-ordered list of the rationals. Eh? What do you mean? It is easy to construct a well-ordered list of the rationals. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Eckard Blumschein on 18 Apr 2005 09:17 On 4/18/2005 2:01 PM, Dik T. Winter wrote: > In article <42639096.600(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > ... > > > Please name one that is not. I'll admit that some are not remedied in a > > > way that is entirely satisfying, but if the paradox is eliminated, where > > > is the superficiality? > > > > E. g., AC does not provide access to a well-ordered list of the rationals. > > Eh? What do you mean? It is easy to construct a well-ordered list of > the rationals. I beg your pardon. In a hurry, I confused the words real and rational. E.
From: Will Twentyman on 18 Apr 2005 14:44
Eckard Blumschein wrote: > On 4/16/2005 11:27 PM, Will Twentyman wrote: > > >>>>but >>>>his fundamental concepts were sound >>> >>>Quantifying a quality is not a sound concept. >> >>We do it all the time. Both ground beef and steak have the quality >>"tasty", yet I can quantify which is *more* tasty. One way is to >>establish a price/kg(or pound). Quantifying the qualities "useful", >>"desireable", etc are the basis of economics. > > Well, my expression was not properly choosen. I shoud better have chosen > "such a quantity" instead. The quality "infinite" is > an either-or-quality, something that cannot be quantified like pregnant, > dead, entirely, countable etc. My point was that, given sufficient motivation, almost anything can be quantified. I can certainly quantify "pregnant" by the trimester, month, etc. I can quantify "dead" by the percent of dead cells/organs, or the number of hours since death. It is sometimes a matter of the *will* to quantify than anything else. Of course, any such quantification is more likely to be accepted if it is reasonable and/or useful. > Different levels of size of infinity are nonsense. And yet Cantor found a way to quantify that very thing. You may wish he hadn't, but he did. > The notion of infinite numbers is self-contradicting. Yet he found a representation for them that is not. > Cantor felt entitled to create a nonsense-mathematics. This is nothing > uncommon at all. However, after more than a hundred years it is > legitimate to ask for at least any tiny benefit. So far I only got aware > of trouble. Then don't use it and don't worry about it. Just make sure your understanding of mathematics and set theory does have a sound basis. >>>I see most of the paradoxes just superficially remedied. >> >>Please name one that is not. I'll admit that some are not remedied in a >>way that is entirely satisfying, but if the paradox is eliminated, where >>is the superficiality? > > E. g., AC does not provide access to a well-ordered list of the rationals. Noting that you meant reals, that is precisely what it does. Of course, it does not give you an explicitly defined function for the well-ordering, but it doesn't claim to. It just claims there is such a function. >>>>I'll give one example where you are simply wrong: "Cýs infinite alephs >>>>only distinguish between countable and uncountable sets." >>> >>>I am aware that they claim to manage much more. I referred to what they >>>really are able to perform. >> >>They distinguish between more than two classes of sets. > > Different classes were just introduced in order to have a justification > to introduce them? I would say, rather, that the different classes are a consequence of the various alephs. >>>Given it would be make sense to subdivide the class of uncountable sets. >>>Do you expect these subdivition nearly equally fundamental as the >>>division into countable and uncountable numbers? >> >>Yes. What is not fundamental about taking the notion of dividing >>infinite sets into the classes "countable" and "uncountable", and then >>repeating the process on "uncountable" to give aleph_1 and aleph_2, then >>aleph_3, then aleph_4...? > > Already the words countable and uncountable mutually exclude each other. > I do not see any sense in this formal repetition that contradicts to > the notion of oo. 130 years of missing benefits seems to confirm my doubt. Missing benefit to who? If they are still around, someone must have seen some benefit. > The fact that the basic process can be > >>repeated is nearly as fundamental as the initial split between countable >>and uncountable. > > > Well. It must not be considered a brilliant idea. It is rather stupid. > Nonetheless, it lacks any justification. Yet it was an idea you would not have come up with. It was radical enough to cause a great deal of controversy. Are you sure it wasn't brilliant? -- Will Twentyman email: wtwentyman at copper dot net |