Cantor Confusion
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From: Virgil on 12 Dec 2006 14:23 In article <1165920622.983740.14320(a)n67g2000cwd.googlegroups.com>, 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. If q_0 is any positive rational number with (q_0)^2 > 2 then the sequence defined by q_(n+1) = (q_n + 2/q_n)/2 is a strictly monotone decreasing sequence of rationals converging to sqrt(2) and p_n = 2/q_n is a strictly monotone increasing sequence of rationals converging to sqrt(2). So that, for each n,the open interval i_n from p_n to q_n contains sqrt(2) and sqrt(2) is the ONLY real number in all of these intervals
From: Virgil on 12 Dec 2006 14:27 In article <1165921167.181546.44810(a)16g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Michael Stemper schrieb: > > > In article <1165615427.980137.7670(a)f1g2000cwa.googlegroups.com>, mueckenh > > writes: > > >William Hughes schrieb: > > > > >> We can now define bijections on potentially infinite sets > > > > > >Only if we consider them being actually infinite. > > > > Would you please define "actually infinite" or "actually infinite set"? > > Actual infinity is the infinity used (or rather claimed to be used) in > every kind of set theory. That hardly constitutes a definition. > > > > > But that would > > >exclude them from being potentially infinite. > > > > Would you please define "potentially infinite" or "potentially infinite > > set"? > > A set or sequence like (n) is potentially infinite, if n can surpass > any upper bound. Then the set of elements in the actually infinite set of naturals numbers in, say, ZFC or NBG, is also potentially infinite? In that case we have no need of only potentially infiniteness at all, as actual infiniteness, as required in ZFC ad NBG, covers it.
From: Virgil on 12 Dec 2006 14:36 In article <1165923504.410525.226600(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > > Please give me all the bits of 1/3. Then I will show the bijection. > > > > > > > > I can't so you can not show a bijection. > > > > > > Representations (= paths) which do not exist cannot be part of a > > > bijection. > > > > So you can not show a bijection (in your opinion), but nevertheless you > > state that you have given a surjection. Do you not think you are > > contradicting yourself a bit? > > I give a surjection on all existing paths. But what has been surjected by WM were not the individual nodes or branches, as required, but /infinite sequences/ of nodes or /infinite sequences/ of branches. Which is not at all the same thing. One can surject the set of infinite sequences of naturals to the set of reals. But that does not prove that the set of naturals and the set of reals biject.
From: Virgil on 12 Dec 2006 14:47 In article <1165928946.144622.162570(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > 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. > > > I do not understand why you always talk about how many lines there are. > > > > Because there are three steps needed to reverse > > the quantifiers: > > > > 1. break up the original statement, giving > > on statement per line > > > > 2. reverse the quantifiers in each statment > > > > 3. recombine the statements to make a single > > statement. > > > > Step 3 (which you have not shown to be possible) depends on > > how many lines there are and whether there is a last line. > > There is only *one* important statement: > > "Every member of the diagonal is a member of at least one line." It is equally true that every member of the diagonal, except one, is NOT a member of at least one line. > > It is not necessary to break up anything and to recombine anything. The > fact that every line is finite guarantees that either the diagonal is > finite too or that the diagonal has at least one element which is not > in the line. Which line? In ZFC and NBG, for every line there is an element of the diagonal not in that line, but there is no element of the diagonal that is not in some line. For x and y in N in ZFC or NBG: (Ax Ey (x < y)) but not (Ey Ax (x < y))
From: Virgil on 12 Dec 2006 14:57
In article <1165930018.676311.91620(a)j72g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > Your statement that the union of all natural numbers is more than any > natural number is simply wrong. It is simply and axiomatically right in ZFC and NBG. And until WM or others can find any internal contradictions in ZFC or NBG, it will stay right. Not that assumption outside of ZFC or NBG are irrelevant in trying to scuttle them. > And you could know it by the fact, that > you cannot name an element of the union which is in no natural number. > According to extensionality then there is no difference. For every natural there is an element in the union of all naturals to in THAT natural. WM's quantifier dyslexia striked again. > > > > - {1,2,3 ...} is not an initial segment > > If no actual existence is assumed, what is the meaning of "..."? If actual existence is assumed, as in ZFC or NBG, the meaning is "all the rest of them". > You can always find a natural number which is not an element of a given > line. But by naming this number you name the line to which it belongs. > It would beas wrong to state that a line containing every natural > number exists, as it would be wrong to state that every natural number > exists. So WM would rather have them come into existence as needed but until then not exist? > But if the diagonal existed for every natural number, then a > line containing every natural number had to exist too. That falsehood is direct evidence of WM's logical corruption. For x and y in N in ZFC or NBG: (Ax Ey (x < y)) but not (Ey Ax (x < y)) > > Therefore the diagonal (= set of all natural numbers) does not exist. Except that it does in both ZFC and NBG. |