Obections to Cantor's Theory (Wikipedia article)
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From: malbrain on 29 Jul 2005 12:30 Han de Bruijn wrote: > Martin Shobe wrote: > > > On Thu, 28 Jul 2005 15:56:32 +0200, Han de Bruijn > > <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>But without the claim that it > >>is the one and only foundation possible. Why not have _several_ pillars > >>that provide a foundation, instead of just one? > > > > Actually, I don't have a problem with that. In a sense, we have that > > now with set theory and category theory (And lets not forget logic). > > But while physics will continue to provide inspirition to mathematics, > > it will not qualify as a foundation for methematics. > > And I don't want that either. Read my lips: > > A little bit of Physics would be NO Idleness in Mathematics > Yes, that's one of the DIAGNOSTICS for SYSTEMS is to provide them with invalid/impossible INPUTS and measure the OUTPUT RESPONSE. karl m
From: malbrain on 29 Jul 2005 12:41 Han de Bruijn wrote: > malbrain(a)yahoo.com wrote: > > > Since I am a materialist: mathematics is created. > > Well, maybe it is both discovered as well as created. No, someone introduces it to you from the outside. Then you agree with its axioms. It's that simple. karl m
From: malbrain on 29 Jul 2005 13:11 Virgil wrote: > In article <52hGe.804$Zh.692(a)tornado.rdc-kc.rr.com>, > "Poker Joker" <Poker(a)wi.rr.com> wrote: > > > "Virgil" <ITSnetNOTcom#virgil(a)COMCAST.com> wrote in message > > news:ITSnetNOTcom%23virgil-3BF47C.00514828072005(a)comcast.dca.giganews.com... > > > In article <MPG.1d51c8215386c4f2989fd9(a)newsstand.cit.cornell.edu>, > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > >> Daryl McCullough said: > > >> > Tony Orlow (aeo6) wrote: > > >> > > > >> > >Robert Low said: > > >> > > > > >> > >> OK, so how many elements are there in the set of all finite > > >> > >> natural numbers? > > >> > >> > > >> > >Some finite, indeterminate number. You tell me the largest finite > > >> > >number, and that's the set size. > > >> > > > >> > So you really think that there is some number n such that n is > > >> > finite, but if you add 1 you get an infinite number? > > >> (sigh) This is the last time I answer this question > > >> NOOOOOOOOOOO!!!!!!!!! > > > > > > But since every natural has an immediate successor, and except for the > > > first. an immediate predecessor, and there are no gaps( at least if one > > > accepts Peano), the only way of getting from finite to infinite is by > > > adding 1 to some finite natural to get an infinite natural. > > > > > >> > Maybe it's 7? Maybe 7 is the largest finite number, and 8 is > > >> > actually infinite? > > >> Don't be stupid. > > > He is just trying to come down to your level, TO. > > >> > > > >> > In fact, a set is finite if and only if the number of elements is > > >> > equal to a natural number. There is no largest natural number, and > > >> > there is no largest finite set. The collection of all finite > > >> > natural numbers is an infinite set. > > > > > >> The set of all finite numbers up to a given number has that number in > > >> it, which is also the set size. Any subset of N has a size that is in > > >> N. > > > > > > What member of N is the size of N? How about the size of N\{1}? The size > > > of the set of even naturals or the size of the set of odd naturals? > > > For the standard theory these all have trivially easy answers, and none > > > of the sizes are members of N. For TO's theory it depends on how his > > > medicatins are affecting him that day. > > > > What member of N is the color of N? How about the color of N\{1}? The > > color of the set of even naturals or the color of the set of odd naturals? > > TO says above, and I quote, "Any subset of N has a size that is in N." > So TO raises the issue of whether certain "sizes" are in N or not. > > As far as I can see, PJ is the first and only person to mention color in > connection with properties of sets. So perhaps PJ should be the one to > answer his own questions, seeing that he seems to be the only one whom > they interest. Actually, Virgil, coloring trees and their parts is a useful concept in graph theory. karl m
From: malbrain on 29 Jul 2005 13:18 Martin Shobe wrote: > On Thu, 28 Jul 2005 15:56:32 +0200, Han de Bruijn > <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > >Martin Shobe wrote: > > > >> 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. > > > >Keep trying. Give up mathematics for a few moments and use what we call > >"common sense". Put your feet on the ground. Just once. Please ... > > Okay. Still doesn't make sense. How can abstractions and > idealizations be physical? And why should "more physical" be better? What do you think that story telling is? A waste of time? Non physical? Every read Homer? karl m
From: Robert Low on 29 Jul 2005 14:13
Robert Kolker wrote: > Han de Bruijn wrote: > >> It's impossible to explain to them that GR and QM only have _refined the >> limits_ wherein classical physics has to be applied. GR and QM _didn't_ >> "replace" classical physics. Not at all. Ask any expert in Computational >> Fluid Dynamics (CFD) and maybe you'll gain some understanding. > Classical electrodynamics cannot account for the properties of > semi-condictors. The anomalous precession of planets falsify Newtonian > gravitation. Galilean Invarant physics is just plain wrong. It is > empirically falsifiable. Now, you're making me agree with Han about something. Roughly speaking, what the better theories have done is to tell us when the classical physics stops being a good model, and when we have to use the better. For a lot of purposes, classical physics is adequate, and we use it (muttering under our breath that it's just because the right theory reduces to this in the limit we're working in). Yes, there are phenomena which we just can't use classical physics to explain: semiconductors are a good example. Superconductors even better. But the gravitation is less clear. You can always invoke other effects---sometimes it works, and you find a planet you didn't know about, and sometimes it doesn't (because Vulcan isn't there) and you have to come up with another explanation. Eventually, it gets to the point where it is more satisfactory (in an entirely non-rigorous sense) to use GR than to try to keep saving the phenomena of Newtonian gravity. And so on. You can probably tell from this, I'm not a big fan of naive falsificationism :-) |