Cantor Confusion
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From: Dik T. Winter on 16 Oct 2006 22:48 In article <1161019210.489120.290720(a)e3g2000cwe.googlegroups.com> "MoeBlee" <jazzmobe(a)hotmail.com> writes: > mueckenh(a)rz.fh-augsburg.de wrote: > > A good, if no the best source to learn about the different meanings of > > infinity would be Cantor's collected works. > > Set theory has advanced since Cantor. The best source to learn about > current set theoretic definitions of 'infinite' is not Cantor. Indeed. There are some statements in that work that are false according to current set theory (as I already did notice before). The works are a good starting point when you want to talk about the history of set theory, but that is all. One of the most serious errors can he found in the statement that "to count sets of first cardinality you need ordinals of the second class" where (I may have first and second wrong), first cardinality means (in current terminology) aleph-0 and second class ordinals means (in current terminology) omega and larger (until omega^omega or somesuch). The error is of course that there is a set of cardinality aleph-0 that can be counted using finite ordinals only (the natural numbers). It is exactly the same error that is present in Tony's presentations and in many of Wolfgang's presentations. In spite of Wolfgang's protestations, the contents of the set N represent a potential infinity while the size represents an actual infinity. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 16 Oct 2006 22:58 In article <1161027796.150952.52530(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1160933229.072292.316580(a)e3g2000cwe.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > > > Dik T. Winter schreef: > > > > In article <1160856895.115824.134080(a)b28g2000cwb.googlegroups.com> > > > > Han.deBruijn(a)DTO.TUDelft.NL writes: > > ... > > > > > Come on, guys! You all know that, in the world of approximations, > > > > > 2 _is_ the square of a rational and the circle _is_ squared. > > > > > > > > I thought you were talking mathematics? > > > > > > I thought approximations were a part of mathematics? > > > > They are. Numerical mathematics in particular. But also in them, 2 is > > *not* the square of a rational number. The best you can state is that > > there is a rational number whose square approximates 2 with a certain > > precision. > > Correct. Doing better is impossible. In numerical mathematics. In actual mathematics you can state that sqrt(2) squared is exactly equal to 2, and you can prove that it is not rational. I have no idea why you have problems with this. Stuff like the square roots of integers have been used for quite a long time in useful mathematics (like factorising integers, to defeat encryption methods). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 16 Oct 2006 23:05 In article <1161027917.596265.118160(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1160858083.663838.195390(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > hence not existing, hence the > > > "if there are actually infinitely many positions" contradicts itself. > > > > Pray, first show a *valid* mathematical proof of your statement above. > > It is impossible to show a "valid" mathematical proof against set > theory. Ah, so you agree that you can not prove an inconsistency using mathematical terms of proof. > We have discussed the vase and I would not have believed in > advance that anybody could maintain arguments here like Virgil and > William and others. No. You never would believe that anybody would use mathematical proofs against your intuition. So be it. The difference is that they provide a mathematical model for the problem, which you have not given at any time. > Therefore I am sure set theory will never be > contradicted --- its proponents simply will die out. That is the way of nature, and successors will fill the voids. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 16 Oct 2006 23:10 In article <1161028138.924247.239940(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > The set of bits is determined: exactly 100. What you can build from 100 > > > bits belongs to the power set of this set. It is probably large but > > > certainly not infinite. > > > > The set of bits is determined. The set of numbers you can build from it is > > indeterminate. But whatever way you build your set of numbers, the size is > > certainly <= 2^100, > > The cardinal number of any such set is =< 100 Redo your calculations. With 100 bits there are 2^100 possibilities, so the cardinality for the set of numberss represented is <= 2^100. > > and whatever way you build your set of numbers, there > > is a largest one that can be determined. > > But nobody knows it. That is just the case with N, only a bit more > sophisticated. In what way? In the case of N it is known how the numbers are built up. > > Oh, perhaps. What is the relevance to mathematics? > > Nothing, as soon as we withdraw to call set theory mathematics. Oh. In that case, please do not call it mathematics. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: imaginatorium on 17 Oct 2006 00:42
David Marcus wrote: > stephen(a)nomail.com wrote: > > What does it mean for a thing in the real world to be true? > > How do you know if a thing in the real world is true? > > > > Consider the twin slit experiment. Is the fact that none of > > the following accurately describe the situation an inconsistency? > > a) the photon goes through one slit > > b) the photon goes through both slits > > c) the photon goes through neither slit > > In Bohmian Mechanics (and similar theories), the photon goes through > only one slit. Physicists could learn something about logical thinking > from mathematicians. But isn't it true that while "undoubtedly" going through one slit, it interferes with itself "undoubtedly" not going through the other slit? Brian Chandler http://imaginatorium.org |