Obections to Cantor's Theory (Wikipedia article)
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From: Martin Shobe on 28 Jul 2005 07:45 On Thu, 28 Jul 2005 10:04:43 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >Robert Kolker wrote: > >> Sets are not physical. They are abstract. Search the world over and you >> will not find a set. In the real world trees exist but forests do not. >> Forests exist in our heads. > >If I say to you that I can draw a straight line, will you then respond >to me that straight lines cannot be drawn, because they are "abstract"? Of course they can't be drawn. Lines do not have ends, but I'm pretty sure that what you draw will have two of them. In addition, lines have no width, I'm pretty sure that what you draw will have a width, (although this width will vary with where and when you are). Lines are straight, I'm pretty sure your line won't be. Martin
From: Han de Bruijn on 28 Jul 2005 07:50 Martin Shobe wrote: > And the gist of our argument is that not being able to "go back to > earth" is not a problem. Mathematics is not restricted to those > things that can "go back to earth". Precisely. But perhaps "_useful_ mathematics" is restricted to those things that can "go back to earth". If we make a distinction between useful and useless mathematics, formalize it and present that to our politicians, tax payers and bosses. Would that be a problem? No? Just as a matter of prevention, shouldn't we at least _try_ to find a way of living together, then? Han de Bruijn
From: Dik T. Winter on 28 Jul 2005 07:43 In article <1122503371.218414.268340(a)g47g2000cwa.googlegroups.com> malbrain(a)yahoo.com writes: > Dik T. Winter wrote: .... > > > > > The C language is defined by the C standard, as defined by ISO. > > > > > There are no "unbounded" standard types in the C language. karl m > > > > > > > > Who is talking about C? > > > > > > Of the billions of computer systems deployed since the micro-computer > > > revolution, the overwhelming majority are programmed with C. > > > > That is not an answer. > > Well, the OBVIOUS answer to your question is, "I'm talking about C" > However, I'm not that vulgar. I tend to translate discussions into C > because I find it to be more universally understood than java. karl m So what? When someone talks about java with "unbounded" standard types, what is the point stating that C does not have "unbounded" standard types? -- 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 de Bruijn on 28 Jul 2005 07:54 Martin Shobe wrote: > On Thu, 28 Jul 2005 10:04:43 +0200, Han de Bruijn > <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >> >>If I say to you that I can draw a straight line, will you then respond >>to me that straight lines cannot be drawn, because they are "abstract"? > > Of course they can't be drawn. Lines do not have ends, but I'm pretty > sure that what you draw will have two of them. In addition, lines > have no width, I'm pretty sure that what you draw will have a width, > (although this width will vary with where and when you are). Lines > are straight, I'm pretty sure your line won't be. Preface to Isaac Newton's Principia: "the description of right lines and circles, upon which geometry is founded, belongs to mechanics". Han de Bruijn
From: Martin Shobe on 28 Jul 2005 08:07
On Thu, 28 Jul 2005 11:48:09 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >Martin Shobe wrote: > >> Since sets aren't physical, it is not physically correct to assert >> that a e A ==> a c A. From what I understand of the history of set >> theory, physics wasn't the inspiration, and it certainly isn't its >> purpose at the current time. > >Since sets aren't physical, since straight lines aren't physical, since >numbers aren't physical. Of course they _are_ physical. Or better: they >_were_ physical. Though some have been more physical than others. Those >which have been more physical are the better ones, i.e. straight lines. I'm sorry, but I can't make any sense out of this. >> That doesn't mean that you can't explore theories where a e A ==> a c >> A. Just don't confuse what you are doing with standard set theory. >> (And it would be nice if you indicate that you aren't doing standard >> set theory, if for no other reason than to avoid some confusion.) >Heh, heh. That's precisely the first thing I want: throw _doubt_ upon >everything that everybody finds so certain. And I thought that I have >indicated that well enough, all over the place, ever since 1989. But your objections don't cast any doubt, because they are irrelevant to what you are objecting to. Set theory does not have physics as it's inspiration, so failing to satisfy some physics viewpoint is irrelevant. And what you do in non-standard theories doesn't necessarily impact the standard theories. And this is the first time I've seen you indicate that you were talking about non-standard set theories. It had always appeared to me that you wished your conclusions to apply to ZFC, etc. >> If you are going to use that as an axiom, then the set theory you add >> it to should include sets that aren't well founded. Make that very >> few well-founded sets. > >As a consequence, mathematics can no longer be founded on set theory. >That's precisely what I want: set theory as a relatively _unimportant_ >branch of mathematics. Different meanings of founded. A "well-founded set" is a set that doesn't have an infinite decent on e. (The inclusion operator). It might be possible to found (make rigourous) the rest of mathematics using only a few sets that aren't well-founded (infinite decent on e). I don't know for certain though. If you want to make set theory an unimportant branch of mathematics, your best bet is to find an alternative foundation that is acceptable to mathematicians. (I.e. phsyics doesn't cut it). The only real competitor for foundations of mathematics that I'm aware of is category theory, and that does it by providing a foundation for set theory, and using that to found other areas. Martin |