Obections to Cantor's Theory (Wikipedia article)
in [Logic]
|
Prev: Derivations
Next: Simple yet Profound Metatheorem
From: stephen on 26 Jul 2005 02:19 In sci.math malbrain(a)yahoo.com wrote: > stephen(a)nomail.com wrote: >> In sci.math malbrain(a)yahoo.com wrote: >> > Barb Knox wrote: >> >> In article <1122338688.718048.162860(a)g47g2000cwa.googlegroups.com>, >> >> malbrain(a)yahoo.com wrote: >> >> >> >> >Barb Knox wrote: >> >> >> In article <MPG.1d4ecd45545679a8989f6b(a)newsstand.cit.cornell.edu>, >> >> >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >> >> >> [snip] >> >> >> >> >> >> >keep in mind that >> >> >> >inductive proof IS an infinite loop, so that incrementing in the loop >> >> >> >creates >> >> >> >infinite values, and the quality of finiteness is not maintained over those >> >> >> >infinite iterations of the loop. >> >> >> >> >> >> Using your computational view, consider the following infinite loop >> >> >> (using some unbounded-precision arithmetic system similar to >> >> >> java.math.BigInteger): >> >> >> >> >> >> for (i = 0; ; i++) { >> >> >> println(i); >> >> >> } >> >> >> >> >> >> Now, although this is an INFINITE loop, every value printed will be >> >> >> FINITE. Right? >> >> > >> >> >Not so fast. The behaviour of incrementing i after it reaches INT_MAX >> >> >is undefined. >> >> >> >> Not so fast yourself. You missed the part about "unbounded-precision >> >> arithmetic system", which has no max. >> >> > Sorry, but the C standard admits no such system. INT_MAX must be >> > declared by the implementation. There is no room for exceptions. karl >> > m >> >> Why are you talking about C? > Because C is "better" defined than java, and the example is written in > C. karl m The example is not written in C. It is perfectly legal Java code, and C++ code, and C# code, and Javascript. Given that 'println' is not a standard C function, but it is a standard Java function, and the author identified the example as Java, it is pretty clear the example was written in Java. Stephen
From: Han de Bruijn on 26 Jul 2005 03:19 Chris Menzel wrote: > On Thu, 21 Jul 2005 09:57:21 +0200, Han de Bruijn > <Han.deBruijn(a)DTO.TUDelft.NL> said: > >>Virgil wrote: >> >>>Every bit of "Cantorianism" has been well enough defined for the >>>understanding of thousands upon thousands of people. That TO fails where >>>so many have succeeded says more about TO than about the adequacy of >>>"Cantorianism's" explanations. >> >>The fact that a faith has millions of adherants doesn't say anything >>about its validity. It says something about the society wherein it is >>accepted, though. > > Faith?? "Cantorianism" is embodied in a completely rigorous axiomatic > theory. The propositions you find unacceptable are demonstrably valid > in that theory. There is not a lick of faith involved. Instead of > tossing off idiotic comparisons to religious belief -- an inevitable > rhetorical haven for cranks and crackpots -- you might consider a > genuinely mathematical response: point out the axiom(s) of set theory > you consider unacceptable and defend your rejection of them with > arguments; or simply embark straightaway on the development of an > equally rigorous alternative. Responses like yours only show you > haven't the least clue what mathematics is. Yeah. Sure. It's all so rigorous that they don't even have a clue how to handle an elementary limit which is the easiest one I've ever done. See elsewhere in this thread. I can't believe my eyes. Han de Bruijn
From: Han de Bruijn on 26 Jul 2005 03:25 MoeBlee wrote: > georgie wrote: > >>>Faith?? "Cantorianism" is embodied in a completely rigorous axiomatic >>>theory. >> >>How can axioms be rigorous? Aren't they supposed to be self-evident? >>Doesn't that sort of imply that you "believe" them to be true? Isn't >>that faith? > > Axioms are rigorous in that there is an effective method by which to > determine whether a formula is an axiom. Non-logical axioms are, by > definition, true in some models and not true in others. Just to study a > theory, one does not have to commit to a belief that a particular model > is one of the real world, whatever one takes 'the real world' to mean. Huh? And again: huh? Flabbergasted ... Han de Bruijn
From: Han de Bruijn on 26 Jul 2005 03:36 Robert Kolker wrote: > Without set theory there would be no rigorous theory of real or complex > variables. Real and complex variables existed well before the advent of set theory. > Without set theory there would be no point set topology, a > general theory of spaces based on the abstract notion of nearness. Yeah, yeah. And another bunch of theories which are hardly applicable. > Set theory has made modern mathematics what it is today. I cannot deny the latter, alas. Han de Bruijn
From: Han de Bruijn on 26 Jul 2005 03:44
Robert Kolker wrote: > Han de Bruijn wrote: > >> >> The mathematics or the physicists, whoever does it. > > > The empirical content of any mathematically formulated theory of physics > lies in the mapping of the mathematatical theory into the space of > measurables and physical operations. Mathematics, as such, has no > empirical content whatsoever. True. That's why: A little bit of Physics would be NO Idleness in Mathematics Han de Bruijn |