Cantor Confusion
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From: Ralf Bader on 9 Dec 2006 01:38 Eckard Blumschein wrote: > On 12/6/2006 12:55 AM, Ralf Bader wrote: >> Bob Kolker wrote: >> >>> Eckard Blumschein wrote: >>>> an exact numerical representation available. Kronecker said, they are >>>> no numbers at all. Since the properties of the reals have to be the >>>> same as these of the irrationals, all reals must necessarily also be >>>> uncountable fictions. >>> >>> For the latest time. Uncountability is a property of sets, not >>> individual numbers. There is no such thing as an uncountable real >>> number. Nor is there any such thing as a countable integer. Countability >>> /Uncountability are properties of -sets-, not individuals. >>> >>> You have been told this on several occassions and you apparently are too >>> stupid to learn it. >> >> These people (Blumschein and Mückenheim) don't know how words and notions >> are used in mathematics. Blumschein takes a word, e.g. "uncountable", and >> attempts to infer from its everyday usage its mathematical meaning. That >> such-and-such is uncountable means for Blumschein that it doesn't have >> the nature of a number, or something like that. So integers are >> "countable" for Blumschein, whereas irrationals are swimming in that >> sauce Weyl spoke about (but not meaning what Blumschein thinks) and >> therefore are "uncountable" in Blumscheins weird view. For Blumschein, >> your explanations are just your prejudices. It is pointless to repeat >> them. He can't understand you. > > According to the axiom of extensionality, a set has been determined by > its elements. Why do you not admit the possibility that countability of > a set requires countable numbers. Doesn't it make sense? I didn't find a > single counter-example. And I didn't find any counterexample to the fact that anything you write is stupid gibberish. Nonetheless I will give you a "proof" of the inconsistency of sert theory: There are different definitions of real numbers which, however, are said to be equivalent. One is with Dedekind cuts, by which a real number is a set of rational numbers, x = {q e Q | q < x}. This is a countable set (so you may belabor this by rambling about "countable numbers"). Another definition of real number is as a class of Cauchy sequences. A diagonal argument easily reveals the fact that those classes are uncountable. So in this view a real number is a certain uncountable set, and we have a "contradiction": A real number is a countable set and an uncountable set at the same time. Of course, this is utter nonsense. But I think it fits well into your (and Mückenheim's) follies. Ralf
From: David Marcus on 9 Dec 2006 02:12 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > Bob Kolker wrote: > > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > >>> You cannot imagine the integer [pi*10^10^100]. > >> > >> That is not an integer, dummkopf. It is an irrational real number. > > > Dummkopf? Who? Doesn't "[ .. ]" stand for the "floor" function? > > > Han de Bruijn > > Not anywhere with which I am familiar. The more usual notation is to use brackets without the top horizontal lines. Not an ASCII character. -- David Marcus
From: David Marcus on 9 Dec 2006 02:16 Han de Bruijn wrote: > Lester Zick wrote: > > On Thu, 7 Dec 2006 03:00:21 -0500, David Marcus > > <DavidMarcus(a)alumdotmit.edu> wrote: > > > >>If you use ZFC (or something similar) as your foundation for > >>mathematics, then everything is a set. Of course, while solid > >>foundations are good to have, if you are living on an upper floor, you > >>may prefer to ignore what is going on in the basement. > > > > So you're saying that set "theory" is all of mathematics? Of course > > since what you say isn't necessarily true that's not exactly a ringing > > endorsement of set "theory". > > It's quite simple. Set Theory can not be the foundation for mathematics, > because NOT EVERYTHING IS A SET. E.g. a calculation is mathematics, but > it's not a set. Set theory may be of limited use, but it's supremacy is > complete nonsense, and will be overruled in time. Are you trolling or are you serious? I suspect you are serious. If so, the first thing you should realize is that the words "foundation" and "supremacy" are about as close to each other as are "basement" and "penthouse". -- David Marcus
From: David Marcus on 9 Dec 2006 02:30 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > Lester Zick wrote: > >> On Thu, 7 Dec 2006 03:00:21 -0500, David Marcus > >> <DavidMarcus(a)alumdotmit.edu> wrote: > >> > >>>If you use ZFC (or something similar) as your foundation for > >>>mathematics, then everything is a set. Of course, while solid > >>>foundations are good to have, if you are living on an upper floor, you > >>>may prefer to ignore what is going on in the basement. > >> > >> So you're saying that set "theory" is all of mathematics? Of course > >> since what you say isn't necessarily true that's not exactly a ringing > >> endorsement of set "theory". > > > It's quite simple. Set Theory can not be the foundation for mathematics, > > because NOT EVERYTHING IS A SET. E.g. a calculation is mathematics, but > > it's not a set. Set theory may be of limited use, but it's supremacy is > > complete nonsense, and will be overruled in time. > > But everything can be modelled as a set. You simply do not understand > what "foundation" means in this context. Any calculation can > be rewritten as a set theory problem. It would be long, cumbersome, > and impractical, but it could be done. Just as an computer program > can be transformed into a Turing machine. Yes, but I think it is a bit simpler than that. The study of algorithms involves the study of certain functions on certain number systems. Such functions and numbers can be handled in ZFC. That's the real point of the construction of the natural numbers from sets: to show that set theory can be used as the foundation for arithmetic and hence analysis, etc. However, if you are doing numerical analysis or calculus, you think of N, Z, Q, R, C, etc. as primitive objects. You don't care that they can be modelled as sets. There was an interesting thread recently on sci.math that discussed whether Wiles's proof of Fermat's Last Theorem could be carried out in ZFC. It seems that as written it uses some category theory stuff that can't be done in ZFC. I think those in the know said that it seemed likely it could be rewritten to work in ZFC. -- David Marcus
From: David Marcus on 9 Dec 2006 02:42
cbrown(a)cbrownsystems.com wrote: > Han de Bruijn wrote: > > Lester Zick wrote: > > > > > On Thu, 7 Dec 2006 03:00:21 -0500, David Marcus > > > <DavidMarcus(a)alumdotmit.edu> wrote: > > > > > >>If you use ZFC (or something similar) as your foundation for > > >>mathematics, then everything is a set. Of course, while solid > > >>foundations are good to have, if you are living on an upper floor, you > > >>may prefer to ignore what is going on in the basement. > > > > > > So you're saying that set "theory" is all of mathematics? Of course > > > since what you say isn't necessarily true that's not exactly a ringing > > > endorsement of set "theory". > > > > It's quite simple. Set Theory can not be the foundation for mathematics, > > because NOT EVERYTHING IS A SET. E.g. a calculation is mathematics, but > > it's not a set. > > But calculations performed on the real numbers can, in principle, be > translated to operations on sets and back again to obtain identical > results. Yes, but I think it is simpler than that. Anyone who has been reading sci.math must be aware of the model of the natural numbers as sets, i.e., 0 is the empty set, 1 is {0}, 2 is {0,1}, etc. So, the natural numbers can be modelled as sets. We can then construct the integers, rationals, and reals from the naturals using the standard constructions. And ordered pairs and functions can be modelled as sets. So, now we can do just about anything mathematical and all we need are sets. > > Set theory may be of limited use, but it's supremacy is > > complete nonsense, and will be overruled in time. > > "Supremacy"? It's not somehow "better" than other branches of > mathematics. The fact that in principle we can perform calculations > equivalently by performing certain set theoretic operations doesn't > mean that the latter is a "good" way of performing calculations. The point of a common foundation is to have as few axioms as possible. For example, we don't need one set of axioms for analysis and another set for geometry. > Set theory is primarily a tool used to unify different branches of > mathematics by providing these different branches with a common > language. Bourbaki had a lot to do with promoting the view that mathematics is about sets and structures on sets. This is somewhat different from using set theory as a foundation. Here it is being used as an organizing principle. I'd rather say "organizing principle" than "unify". A different view is that mathematics is about objects and maps between them. This leads to the idea of using category theory as a foundation. I could be wrong, but I think while it is plausible that this could be done, it hasn't been done in detail. -- David Marcus |