Cantor Confusion
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From: Tony Orlow on 12 Dec 2006 11:03 mueckenh(a)rz.fh-augsburg.de wrote: > Tony Orlow schrieb: > >>>> Perhaps, with a countable number of bits. But, there are more than any >>>> finite number of reals in the unit interval, and this is an infinite >>>> number. >>> Where *are* they? They canot be addressed. They cannot be imagined. >>> They cannot be known. And, moreover, the proof proving their existence, >>> is false. >>> >>> Quite a poor existence. >>> >>> Regards, WM >>> >> Well, the proof is simple. Any finite number of subdivisions of any >> finite interval will only identify a finite number of real midpoints in >> that interval, between any two of which will remain more real midpoints. > > And they are (can be represened by) rational numbers too. That's fine. If you don't like irrationals, ignore them. There are an actually infinite number of rationals in a unit interval. > >> Therefore, there are more than any finite number of real points in the >> interval. Of course, most of them require infinite strings of bits to >> specify, but that doesn't mean the point doesn't exist. > > Tony, have you ever wondered why physics is atomistic but mathematics > is (assumed to be) not? > > Regards, WM > Physics used to be more continuous, but atoms and quantum effects have been discovered. Time and space may even be discrete. Mathematics can reflect that, or treat things as continuous. I don't think we've determined for sure that nothing is continuous. Do you?
From: William Hughes on 12 Dec 2006 13:37 Tony Orlow wrote: > Well, the proof is simple. Any finite number of subdivisions of any > finite interval will only identify a finite number of real midpoints in > that interval, between any two of which will remain more real midpoints. > Therefore, there are more than any finite number of real points in the > interval. This just shows that the number of real points is unbounded. It does not show it is infinite (unless of course you use the fact that any unbounded set of natural numbers is infinite). - William Hughes
From: Virgil on 12 Dec 2006 13:41 In article <1165918968.472472.87200(a)80g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > Why should there be a problem as long as *all* lines are finite? > > > > The problem is not with any one line, it is with > > the attempt to combine all lines into one. > > The fact that *all* lines are finite does not > > mean there is a last line. > > But it means that every line is finite. Therefore for every line we can > reverse quantifiers. But not for all lines. > I do not understand why you always talk about how many lines there are. Because that is what is being quantified. > > > > > > Do you have a counter example where quantifier reversal in any finite > > > line / set was prohibited? > > > > No. For any natural number n > > the statement > > > > > > For every m<=n, there exists a line L(m) > > such that m is in L(m) > > > > can be reversed to form > > > > There exists a line L(n) such that for every m <=n, > > m in in L(n) > > > > However the L(n) is different in > > every pair. You can combine all lines into > > > > There exists a line L such that for every natural number n > > n is in L > > > > if and only if there is a last line. > > > The finiteness of all lines would then imply that there are only > finitely many lines (= natural numbers). If your logic were valid, the finiteness of each point on a circle would equally imply that there are only a finite number of points on a circle. > Or the finiteness requirement > of all lines must be dropped. WM deceives himself over the number of lines versus the number of elements in any one line. The number of lines is not finite but the number of elements in any one line is finite. > Then we need a line with infinitely many > elements. WM needs a transfusion of sanity. > I think you should consider it as a fact, that an actually infinite set > of finite numbers is nonsense. While any physical infiniteness maybe impossible, until someone can show that ZFC or NBG is internally contradictory, actually infinite sets will continue to flourish within them. The only way to maintain this claim is > to assert that a diagonal can have more elements than any line. That > alternative is not acceptable. It is to me, and to anyone who does not assume a priori that infinite sets are impossible. > > Regards, WM
From: Virgil on 12 Dec 2006 13:44 In article <1165919243.687026.305870(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > Do you now claim > > > > the natural numbers are not a potentially infinite set? > > N is a potentially infinite set. > > > > the natural numbers do not exist? > > More than enoug do exist. (More than we will ever need could be brought > to existence.) > > > > one can't define functions on the natural number? > > One can define such functions called sequences, but that does not > guarantee that all the pairs do actually exist. The pair (x, f(x)) with > x = floor(pi*10^10^100) does not exist, for instance. If f(x) = 1 for all x then the pair (x,1) exists for all x. > > > > Not that all of the above is completely independent > > of whethe the natural numbers actually exist, we just > > need that the natural numbers exist. > > > > I do not claim that a potentially infinite set actually > > exists, I merely claim that a potentailly infinite set exists. > > That is all I need. > > But what do you mean and understand by "to exist"? In *every* set > theory, starting from Cantor, it means to exist actually. I am not aware of any set theory in which any set is declared to "actually" exist, though there are those in which sets are declared to exist.
From: Virgil on 12 Dec 2006 13:58
In article <1165920355.711390.324330(a)16g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > > This is the argument you present. > > > > > > > > > > > > > > 1) The set of lines is also the potentially infinite set of natural > > > > > numbers. > > > > > 2) Every element of the diagonal is in at least one line. > > > > > 3) Every initial segment of the diagonal is (in) a line. > > > > > > > > No, there is one initial segment of the diagonal that is > > > > not in a line. > > > > > > Then name the element of the diagonal please, which supports your > > > claim. > > > > > > There is one initial segment of the diagonal which is not defined > > by a single element. > > > > The potentially infinite sequence > > > > {1,2,3 ...} > > > > is an initial segment of the diagonal, but it is not in a line, nor > > does it have a largest element. > > If it is not in any line then there must be at least one element of it > which is not in any line. Not quite. In order to achieve that the diagoal is not in any linem all that is required is: Given any line there is an element of the diagonal not in THAT line. It is not requires that: There is an element of the diagonal that is not in any line. > If you say it exists, then you must say by > what element it differes from any finite initial segment. By the next element after that finite segment, obviously. > > > > (It is contained in the union of all lines, but the > > union of all lines is not a line) > > That is a void assertion unless you can prove it by showing that > element by which the union differes from all the lines. Not quite. In order to achieve that the diagoal is not in any linem all that is required is: Given any line there is an element of the diagonal not in THAT line. It is not requires that: There is an element of the diagonal that is not in any line. Failure to recognize the distinction between the two is another instance of WM's pervasive QUANTIFIER DYSLEXIA. For S = the set of all naturals, or for S = the set of all reals, or for S = the set of all rationals between 0 and 1, and for lots of other ordered sets, it is true that (Ax in S) (Ey in S) (y > x) but it is false that (Ey in S) (Ax in S) (y > x) Conflation of the two, as WM is in the habit of doing, is called QUANTIFIER DYSLEXIA. And WM has a serious case of it. |