Cantor Confusion
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From: Bob Kolker on 9 Dec 2006 14:50 David Marcus wrote: > Bob Kolker wrote: > >>David Marcus wrote: >> >> >>>Nonsense. If you use a word, you must define it. Please define >>>"countable number". >> >>Not every word can be (verbally) defined. This would lead to >>circularity. Some words can only be defined by ostention or example. > > > Or, by giving axioms that the word satisfies. However, something must be > given or we are only discussing poetry. The axioms provide the semantics. This is a form of ostention. It is not as concrete as point to particular examples as a different set of meanings could satisfy the axioms. Bob Iolker >
From: Han.deBruijn on 9 Dec 2006 14:57 stephen(a)nomail.com schreef: > 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. > > It's not much of a case. You have not presented any evidence that there exists > any sort of mathematics not describable by set theory. Until such evidence > exists, the hypothesis that mathematics can be modelled with set theory has > not been falsified. And don't bother presenting something that uses limits, > functions, etc. as all of those things can be modelled with set theory. Ah, now you are trying to put the burden on me. But that is false play, of course. You said something like "all heat is phlogiston". I do not have to argue that this is not so. The burden remains yours. I didn't even say that set theory is useless within mathematics. I've only said that there's more to mathematics than set theory. And I find the claim that set theory is the one and only foundation of mathematics merely a manifestation of a narrow mind, if not irresponsible behaviour. One shouldn't narrow down a beautiful discipline to such a poor paradigm. To those who are unable to see it, there is no evidence that there is any sort of mathematics not describable by set theory. But this is a vicious circle. Mainstream mathematics simply DOES NOT ALLOW any sort of mathematics that violates set theory. That's what the whole thread here is about. And, as you know, there are, and there have been, many other such threads in 'sci.math'. Personally, I find that (infinitary) set theory is full of inconsistencies and paradoxes. It's unbelievable that it's still finds so much support. Must be on the wrong planet .. Apart from this, Im currently in the process of posting some pieces of mathematical theory on Chebyshev polynomials. I seldomly feel the need to employ set theory in this work (I've done it accidentally, though). The reason is that set theory, as such, contributes virtually nothing to understanding. I can do quite well without it, most of the time. Han de Bruijn
From: David Marcus on 9 Dec 2006 15:17 Han.deBruijn(a)DTO.TUDelft.NL wrote: > stephen(a)nomail.com schreef: > > > 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. > > > > It's not much of a case. You have not presented any evidence that there exists > > any sort of mathematics not describable by set theory. Until such evidence > > exists, the hypothesis that mathematics can be modelled with set theory has > > not been falsified. And don't bother presenting something that uses limits, > > functions, etc. as all of those things can be modelled with set theory. > > Ah, now you are trying to put the burden on me. But that is false play, > of course. You said something like "all heat is phlogiston". I do not > have to argue that this is not so. The burden remains yours. I didn't > even say that set theory is useless within mathematics. I've only said > that there's more to mathematics than set theory. All the mathematics I've ever seen in a math class or read in a math book or journal or done myself can be done in ZFC. Admittedly, there are large areas of mathematics that I know little or nothing about. Still, that would seem to be quite a bit of evidence right there for the statement that ZFC can be used as a foundation for mathematics. So, the burden is now on you to show some mathematics that can't be done in ZFC. > And I find the claim > that set theory is the one and only foundation of mathematics merely > a manifestation of a narrow mind, if not irresponsible behaviour. That is a straw man. Who has claimed that? The actual claim is that it is an economical foundation that has been worked out in detail. As such, it serves the purpose of providing a foundation. > One shouldn't narrow down a beautiful discipline to such a poor paradigm. > > To those who are unable to see it, there is no evidence that there is > any sort of mathematics not describable by set theory. But this is a > vicious circle. Mainstream mathematics simply DOES NOT ALLOW any sort > of mathematics that violates set theory. Quite a silly thing to say. Do you have any evidence for such an absurd statement? I can give some contrary evidence: I recall reading an article by Saunders MacLane (in the Notices of the AMS, I think) where he asked the logicians to put more effort into developing category theory as a foundation. He didn't feel that set theory was a natural foundation for algebra. > That's what the whole thread > here is about. And, as you know, there are, and there have been, many > other such threads in 'sci.math'. Personally, I find that (infinitary) > set theory is full of inconsistencies and paradoxes. "Personally"? Claims of inconsistency and paradox are a dime a dozen. Please provide a specific inconsistency. > It's unbelievable > that it's still finds so much support. Must be on the wrong planet .. Yes, you'd probably be happier on Mars. On this planet, we require facts and evidence, not personal assertions. > Apart from this, Im currently in the process of posting some pieces of > mathematical theory on Chebyshev polynomials. I seldomly feel the need > to employ set theory in this work (I've done it accidentally, though). > The reason is that set theory, as such, contributes virtually nothing > to understanding. I can do quite well without it, most of the time. Once again, you demonstrate that you don't know what the word "foundation" means. -- David Marcus
From: Han.deBruijn on 9 Dec 2006 15:23 stephen(a)nomail.com schreef: > functions, etc. as all of those things can be modelled with set theory. The topic of functions has been handled separately on my web page: http://hdebruijn.soo.dto.tudelft.nl/www/grondig/natural.htm#fd In a nutshell: the mainstream definition is narrow-minded because the whole notion of _TIME_ is lacking. Now don't tell me that this is just a side-effect of your process called "abstraction". I would rather name it collateral damage, of a narrow minded paradigm known as set theory. > Nobody can prove the Church-Turing thesis, but that does not prevent > people from being confident that our notion of computability is accurate. It's a good thing that you mention Church. He and others have proposed quite another paradigm than set theory. I would like to call it LABOUR. What do you think about the following: anything in mathematics can be founded on labour ! To say it otherwise: all mathematics is: functions. Ah, and don't say now that functions are sets. Church et al. have been working the other way around, with sensible results, such as functional programming (and languages, like good old LISP). > Nobody can prove anything in science, but that does not prevent people from > placing a lot of confidence in it. For example, there is no proof that gravity acts on > all masses. I am surprised that you do not understand something as basic as that. Absolute rigour is a phantom, even in mathematics. Han de Bruijn
From: Han.deBruijn on 9 Dec 2006 15:58
David Marcus schreef: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > To those who are unable to see it, there is no evidence that there is > > any sort of mathematics not describable by set theory. But this is a > > vicious circle. Mainstream mathematics simply DOES NOT ALLOW any sort > > of mathematics that violates set theory. > > Quite a silly thing to say. Do you have any evidence for such an absurd > statement? [ ... snipped "counter evidence" with Category Theory ... ] Ah, now don't act as if you didn't have those heated debates with some manifest opponents of set theory, i.e. Wolfgang Mueckenheim. I'm not talking about any possibilities to replace set theory by look-alikes. I'm talking about rejecting any monolithic foundation for mathematics, any "foundation" that narrows down my freedom of thinking. I hate any form of NewSpeak, whether it is called Set Theory, Category Theory or Object Oriented Programming. I've seen too many of these. Han de Bruijn |