Cantor Confusion
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From: Bob Kolker on 8 Dec 2006 17:12 Virgil wrote: > > > Those poor homeless cuts still exist as sets. The slickest way of doing Dedickind Cuts is just use the lower half of the cut. A set of rational numbers s such that s has no greatest element and such that every element in Rational - s is greater than any element in s. Bob Kolker
From: Lester Zick on 8 Dec 2006 17:33 On Fri, 8 Dec 2006 15:17:28 +0000 (UTC), stephen(a)nomail.com wrote: >Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> stephen(a)nomail.com wrote: > >>> But everything can be modelled as a set. > >> Define "everything" and prove that claim. > >> Han de Bruijn > >By "everything", I meant everything mathematical. So set "theory" in mathematics is the paradigm for everything mathematical? How circular of you. The problem is you know what you think set "theory" is but you don't know what mathematics is. So whether set "theory" actually represents the paradigm for everything mathematical remains at issue. > Of course that is not 100% precise. Of course it isn't 100% precise but it is 100% circular. >And no, I cannot prove it. Well since you can't prove the truth of anything you say that's not exactly pertinent. > But so far all the various objects of mathematics can be >modelled using set theory. So you say but so you can't prove. But don't let that stop you from saying it again and again and again. > That is what is meant by set theory being a foundation >for mathematics. A foundation for what? So far all set "theory" represents is a foundation for set "theory" and a foundation for extravagant nonsensical claims by set mathematikers they don't even bother to claim are true. > If someone were to invent something "mathematical" (whatever that may >mean exactly) Well the problem is that we're asking you to define the meaning of "mathematical" because you're the one who claims set "theory" as the paradigm for mathematics. > that could not be described in terms of set theory, then set theory would >no longer serve as a foundation. And what evidence do you have that only things described in set "theory" are mathematical? Is this some kind of special intuition? > But given that the basics such as the real numbers, >functions, limits, calculus, etc. all can be founded in set theory, it would have to >be something strange indeed. So what do you imagine is not strange about the things you claim are founded in set "theory"? > Not that there is anything wrong with strange, but you >probably would like it less than set theory. I like set "theory" considerably less than strange. ~v~~
From: Lester Zick on 8 Dec 2006 17:35 On Fri, 08 Dec 2006 16:42:16 +0100, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >stephen(a)nomail.com wrote: > >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> >>>stephen(a)nomail.com wrote: >> >>>>But everything can be modelled as a set. >> >>>Define "everything" and prove that claim. >> >> By "everything", I meant everything mathematical. Of course that is not 100% precise. >> And no, I cannot prove it. But so far all the various objects of mathematics can be >> modelled using set theory. That is what is meant by set theory being a foundation >> for mathematics. If someone were to invent something "mathematical" (whatever that may >> mean exactly) that could not be described in terms of set theory, then set theory would >> no longer serve as a foundation. But given that the basics such as the real numbers, >> functions, limits, calculus, etc. all can be founded in set theory, it would have to >> be something strange indeed. Not that there is anything wrong with strange, but you >> probably would like it less than set theory. > >Correction. By "everything" you probably mean "everything according to >nowadays mainstream mathematics", which _is_, of course, "mathematics", >according to your probably rather limited view. But since you can not >really prove anything of the kind, I will rest my case. Stephen is rather typical of modern set mathematikers. He just makes circular claims regarding set "theory" which he can't prove are true. ~v~~
From: Lester Zick on 8 Dec 2006 17:45 On Fri, 8 Dec 2006 13:07:04 +0000 (UTC), 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. > >> Han de Bruijn > >But everything can be modelled as a set. A set of what? Points? That's preposterous. Nobody doubts everything not only "can be modeled" but actually is a set of predicates.But what makes those predicates points? The problem is not sets as such but the points modern mathematikers like to pretend constitute those sets. > You simply do not understand >what "foundation" means in this context. Any calculation can >be rewritten as a set theory problem. So what? Can set "theory" be rewritten as a set "theory" problem? The problem is how we get the sets in the first place and what the sets are sets of and not what you contend we can do with set "theory" once we get it. > It would be long, cumbersome, >and impractical, but it could be done. Just as an computer program >can be transformed into a Turing machine. So if we allow that TvN mechanics is the paradigm for computable numbers it thereby becomes the paradigm for all mathematics? I suspect you have a considerable number of erroneous hidden assumptions in this little piece of legerdemind. >I do not know what you mean by "supremacy". Do you think 486 >assembly language is the "supreme" programming language? It >currently is sort of a de facto candidate for a foundational >programming language. Macro assembler is a paradigmatic machine programming interface. But there are also better techniques. ~v~~
From: William Hughes on 8 Dec 2006 18:04
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > Recall this post from Dec 1 > > > > We extend this to potentially infinite sets: > > > > A function from the potentially infinite set A to the > > potentially infinite set B is a potentially infinite set of > > ordered pairs (a,b) such that a is an element of A and b is > > an element of B. > > A function, according to modern mathematics, is a set, actually fixed > and complete Yes, but according to modern mathematics the natural numbers are a set, actually fixed and complete. You cannot pick and choose the bits of modern mathematics you want to use. - William Hughes |