Cantor Confusion
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From: Virgil on 1 Nov 2006 14:14 In article <dec9e$454858d1$82a1e228$21143(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <f328c$45470fd4$82a1e228$20321(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>David Marcus wrote: > >> > >>>I guess your definition of "mathematics" is different from mine. > >> > >>Yes. Like your definition of "physics" is different from mine. > > > > Fair enough. > > > > In another post, HdB claimed to have physical evidence that a > > discontinuity would halt time, or something equally idiotic. > > A large mass (as with the balls in a vase close to noon) surely _will_ > halt time, according to the General Theory of Relativity. As those infinitely many balls are required to exist before any are put in the vase, if they are going to stop time at all, they will stop it before any of them are to be put into the vase. So the need for a discontinuity at noon will be no problem. > > > So we are not at all distressed to hear him say that his definition of > > physics is quite different from ours. > > It's Albert Einstein's theory, not mine. Can't you keep up with modern > physics a little bit? I was also unaware that instantaneous displacement of physical objects (balls moving instantaneously from outside a vase to inside, or vice-versa) was possible in conformity with Einstein's Theory of Relativity. Does HdB know of any laboratory where such instantaneous displacements have actually been done? Or even a laboratory which has accessed infinitely many physical objects?
From: Virgil on 1 Nov 2006 14:20 In article <57d64$45485a34$82a1e228$21322(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <bdc92$45476e9e$82a1e228$30478(a)news1.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>It's easy to come up with a correct physical problem and > >>solve it with the wrong mathematics. As you did. > > > > HdB has come up with a mathematical problem which he claims can end the > > world: > > > > HdB claimed that a discontinuity in a mathematical function of time > > causes time to stop. And he claimed this followed from physics. > > > > Thus, if HdB is right, the vase problem will cause the end of the > > world. > > Yes. If it came into _existence_, it would cause the end of the world. > Because an infinite mass would be no less than a Cosmic Disaster. Since it _exists_ only as a gedankenexperiment, a sort of mental game, such fears are irrelevant.
From: Virgil on 1 Nov 2006 14:24 In article <28a6c$45485d1a$82a1e228$21765(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <ei937d$jsr$1(a)news.msu.edu>, stephen(a)nomail.com wrote: > > > >>David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >> > >>>Virgil wrote: > >>> > >>>>In article <bdc92$45476e9e$82a1e228$30478(a)news1.tudelft.nl>, > >>>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>>> > >>>>>It's easy to come up with a correct physical problem and > >>>>>solve it with the wrong mathematics. As you did. > >>>> > >>>>HdB has come up with a mathematical problem which he claims can end the > >>>>world: > >>>> > >>>>HdB claimed that a discontinuity in a mathematical function of time > >>>>causes time to stop. And he claimed this followed from physics. > >>>> > >>>>Thus, if HdB is right, the vase problem will cause the end of the > >>>>world. > >> > >>>Yes, but when? > >> > >>Noon? > > > > As there cannot be any such noon according to HdB's physics experiments, > > it must happen before noon, but after every other time before noon. > > It may seem that you are just making fun. But no, for the first time in > the history of 'sci.math' there's a chance that Stephen and Virgil are > both getting somewhere ... > > Han de Bruijn Where we are getting is sick of HdB's inanities. If one is to play the gedankenexperiment game fairly, there is no possibility of any correspondence with the constraints of physics, so we must say that to play the game at all, physics is outlawed.
From: Virgil on 1 Nov 2006 14:54 In article <1162382183.210161.265840(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1162219026.697699.10850(a)m73g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > It is unbounded but always finite. > > > > Unbounded implies not finite, as finite implies bounded. > > That is the definition of potential infinity. But for this definition > we never have a completet set and never can be sure whether a set is > Dedeking infinite. A set which f(x) = x+1 maps to a proper subset is Dedekind infinite. > > Sorry, to say, but you always mix up these two very different things. A difference which makes no difference is no difference. The only difference in whether the set of finite naturals is potentially or actually infinite is whether one is looking at its members or at the set as a whole. The difference is in what the viewer chooses to see not in what is there to be seen. To those of us who choose to see both the members and the set, the artificial distinction that "Mueckenh" chooses to impose does not exist. > It should be comprehensible that potential infinity is possible without > the axiom of infinity. But that is not what set theory requires. > Therefore it is correct to say that in set theory theory there is no > infinity present or detectable without the axiom of infinity. To claim that there is none present requires proof of absence, which is well beyond WM's capacity to provide. > > WM: > > As long as your strings have a finite number of characters, you have a > > finite number of strings. > > > Virgil: > That may be how things work in in WM's world, but not in mine. > > Since I start, in ZF or equivalent, with at least one actually infinite > > set, |N. I can get as many actually infinite other sets as I wish. > > > If WM claims that his world does not work that way, he is only making > equally unprovable assumptions himself. > > > But WM does not have the power to determine reality, only the power > to > assume that he knows it. > > WM: > The power to recognize it, I hope. A forelorn hope, considering what things he chooses to assume he "knows". > > > WM: > > > > Try to draw the graph of the function f(n) = n. The points lay on the > > > > diagonal of the first quadrant. As long as n is finite, f(n) is finite > > > > too and vice versa. > > Virgil: > > > So what? > > WM: > > This shows that both are finite or both are not. > > > Virgil: > So what? > > The set of even naturals and the set of odd naturals are both finite or > > both infinite, too. But that does not mean that either is finite. > > WM > But you assert that the number f(n) of numbers n is infinite while the > magnitude n of numbers n remains finite. I am not at all sure what WM means by that last bit. The magnitude of the number of naturals up to any natural is finite, but unless there is a last or largest possible natural that is irrelevant to the finiteness/infinitness of the set of all of them. ZF and NBG and other set theories, assume that the set of naturals is not finite even though each of its members is finite. All sorts of people have tried to show that this assumption leads to contradictions within those theories, so far without success. And until someone does succeed in finding an internal contradiction, many, even most, mathematicians will continue to work happily in systems in which there are such infinite sets of finite objects.
From: Virgil on 1 Nov 2006 14:59
In article <1162382347.088624.296000(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1162220492.974057.66580(a)b28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > > > > What kind of well ordering of the reals do you claim to exist? > > > > > > > > A well ordering in which each non-empty subset has a smallest member. > > > > > > That is the definition. > > > > Exactly so! > > > > > > > > > Defined? > > > > > Catalogue? > > > > > List? > > > > > Else? (Please specify) > > > > > > > > A well ordering in which each non-empty subset has a smallest member. > > > > > > > > That's what well orderings are all about. > > > > > > I know. But the definition does not guarantee existence. In particular > > > it does not say how the order in R differs from the order in N or Q. > > > > > > > > If you are asking for a rule for determining which objects come before > > > > others, you should know that no such explicit rule is possible, but that > > > > does not make their existence imposible. > > > > > > What kind of existence do you have in mind? > > > > In a system that has assumed the axiom of choice, the kind that that > > axiom insists on. > > > > Why then do you assert the existence of things which do not exist? If one is working in an axiom system which says they do exist, one goes with the axioms. If WM chooses to work in a different axiom system that is his prerogative, but it is not his prerogative to dictate to me which systems I am allowed to work in. And until WM can provide and internal contradiction to such a system, independent of his own assumptions about it, his criticisms are more religious, being matters of faith, rather than being either logical or mathematical. |