Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 29 Jul 2005 02:18 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.
From: Han de Bruijn on 29 Jul 2005 03:41 Robert Kolker wrote: > Euler and Gauss used "arguments" that would earn an F in modern course work. Then there is definitely something wrong with modern course work. Han de Bruijn
From: Han de Bruijn on 29 Jul 2005 03:44 Robert Kolker wrote: > Han de Bruijn wrote: > >> Preface to Isaac Newton's Principia: "the description of right lines >> and circles, upon which geometry is founded, belongs to mechanics". > > One of Newton's few mistakes. Geodesics are properly defined in > Riemannian geometry. Glad that we have such a bright spirit here that it challenges Newton's. Han de Bruijn
From: Robert Low on 29 Jul 2005 03:49 Martin Shobe wrote: > Maybe I should have said "Set theory does not have physics as it's > immediate inspiration, so failing to satisfy some direct connection to > physics is irrelevant." I was just being picky...and besides, the 'connection to physics' that set theory 'fails to satisfy' is almost entirely obscure to me. I see no physical argument to justify a={a}.
From: Han de Bruijn on 29 Jul 2005 03:55
stephen(a)nomail.com wrote: > No. A set that contains nothing is different than nothing. > An empty box is not nothing. A set that contains nothing is nothing. Since the box is not a part of the set, an empty box is nothing. Nothing but nothingness. > No, there is a problem with how *you* are choosing > to interpret sets. No, there is a problem with how *you* are choosing to interpret sets. > Again, you are insisting that mathematicians see > things your way. It is rather disingenuous of you > to then implore 'why can't we all get along?'. I am insisting that mathematicians learn some common sense. Han de Bruijn |