An uncountable countable set
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From: Virgil on 25 Jul 2006 14:23 In article <1153825745.605304.286610(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Randy Poe schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Of course. Real space is Euclidean or non-Euclidian irrespective of > > > > > the > > > > > axioms which mathematicians prefer. Aithmetic is finite or infinite > > > > > irrespective of the axioms which mathematicians prefer. > > > > > > > > You mean there's something called "arithmetic" which > > > > exists outside mathematics? > > > > > > It is the core of mathematics. > > > > So the core of mathematics is outside mathematics? > > No, but present "mathematics" has little to do with real mathematics. > > Regards, WM Then may God continue to preserve us from the corruption of "mueckenh"'s "real mathematics" and allow us to continue with our own real mathematics.
From: Virgil on 25 Jul 2006 14:30 In article <1153826683.714320.313740(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: > > > Dik T. Winter wrote: > > Mueckenh writes: > > >> But it is obvious for *all* *finite* *unary* number. And only > > >> finite unary numbers are involved in the second list below: > > >> > > >> The *- sum of > > >> 0.1 > > >> 0.11 > > >> 0.111 > > >> ... > > >> 0.111... > > >> > > >> 0.111... is not a finite unary number. Rather, I would say it is not a > > >> unary number at all. > > >> 0.111.. is not a terminating rational number. > > >> [...] > > > > > > > The list: > > 0.1 > > 0.11 > > 0.111 > > ... > > contains only terminating rational numbers (fractions), all of them of > > the form: > > f(n) = sum{i=1 to n} 10^-i, for all n=1,2,3,... > > > > This means that every f(i) contains a finite number (n) of 1 digits, > > and likewise so does f(i+1) that follows f(i). So we conclude that > > the list is composed entirely of finite terminating fractions. > > > > We also conclude, for exactly the same reasons, that the number > > 0.111111... > > is not in the list, because it is not a terminating fraction, and > > because it is not of the form: > > f(n) = sum{i=1 to n} 10^-i > > We know that every digit which can be indexed by a natural number is in > the list. If 0.111... is not in the list, then not every digit can be > indexed. This is simple logic: > > (every digit of n can be indexed) ==> (n is in the list) > not (n is in the list) ==> not (every digit of n can be indexed) Very simple, but false. You are in effect claiming that there are sets which must be members of themselves. The von Neumann model shows this by refusing to require that any ordinal index itself. Each ordinal is the set of all prior ordinals, including omega.
From: Virgil on 25 Jul 2006 14:46 In article <1153829288.635491.96140(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > I need not count to w. If a column contains one 1 then the *+sum is 1. > This is a definition like that of Cantor's list. Not hardly! > Why does the left column of my extended list differ from the right one? The left column is all zeros.
From: mueckenh on 25 Jul 2006 17:08 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> > Franziska Neugebauer schrieb: > >> >> > In set theory. It is full of the actual infinite. > >> >> > >> >> When I ask for "usage of terms" I want you to name contemporary > >> >> papers/books on set thery that mention/use these terms. > >> > > >> > 1976 is too old? > >> > >> Preface, Wolfgang. You are refering to an elucidation in its preface > >> on page 6. How many times does the book mention and use those two > >> terms on the remaing pages? > > > > I am not in possession of the book but have only my notes. > > Get it and count! Because you are ignorant of mathematical content? Read it yourself! > > > But he frequencay of mentionings may be high though unimporant. > > How will you know if you don't have it at hand? Because I read it. > > > The meaning of "infinity" in set theory is clearly the actual one. > > The other way round. Students first get in touch with plain infinite > sets As you might obtain from the quoted statements, these sets are actually infinite. > and then read from your posts about the ancient terms. To them > the new terms which appear are the ancient ones. Hence from modern set > theoretic standpoint one would clearly define: "Actual infinite" simply > means infinite. Of course. A last you got it. > > > Every expert knows this. Compare Jech's statement. Or compare Hilbert > > or Poincaré. That is just the clue, and it has not changed since > > Cantor's times. > > > > Hilbert (1926, S. 167) > ^^^^ Not all he said was wrong. > > > But here is some newer author: Lorenzen: Die endlichen Weltmodelle der > > gegenwärtigen Naturwissenschaft zeigen deutlich, wie diese Herrschaft > > eines Gedankens einer aktualen Unendlichkeit mit der klassischen > > (neuzeitlichen) Physik zu Ende gegangen ist. > > Befremdlich wirkt dem gegenüber die Einbeziehung des > > Aktual-Unendlichen in die Mathematik, die explizit erst gegen Ende des > > vorigen Jahrhunderts mit G. Cantor begann. > > Elucidation not set theory, not logic not maths. Not female logic. Regards, WM
From: mueckenh on 25 Jul 2006 17:11
Dik T. Winter schrieb: > > If *every* digit can be indexed, then I need not put "finite" between > > "every" and "sequence", because every indexing sequence is finite. A > > digit which is not a finite sequence remote from the first digit cannot > > be indexed and cannot be covered. > > Your deleting of "finite" (or "indexing") here is precisely what I said, the > negation of the axiom of infinity. Any indexing sequence is finite. Right. > But the sequence of *all* digits of 0.111... is neither an indexing sequence, > nor a finite sequence. And it is precisely *that* sequence that can not be > covered by a finite natural. Remember: there is a set that contains all > naturals. There is (clearly) no natural that is the largest in that set. > So there is also no last digit in the sequence of digits of 0.111..., where > the digits are indexed by naturals. But every natural is finite. Every digit which can be indexed by naturals is *by definition* in the true list (not the extended list) and every digit which cannot be indexed is not in the true list. So your attempt to explain that a number which is not in the true list nevertheless could be indexed is completely unfounded. > > > > digits of 0.111... can be covered by a natural. (There is no digit by > > > > which 0.111... could be distinguished from every natural.) But 0.111... > > > > is the union of all of its finite sequences, i.e., of the naturals. > > > > > > Yes. And so? > > > > A union of finite sequences is a finite sequence. > > An arbitrary union of finite sequences is not necessarily a finite sequence. > This is similar to: the union of finite sets is a finite set. But this > argues that the union of {1}, {1, 2}, {1, 2, 3}, ... is a finite set. > Consider the elements of those sets as indices to the elements of the > sequences. This is only true when N is finite, but that is the negation > of the axiom of infinity. True is that the elements of n are finite. Hence there union cannot be infinite, because every union of natural numbers is a natural number. And every natural number is finite. You always intermingle size and number (Anzahl). > > > If you believe that not the whole infinte sequence 0.111... can be > > completely covered by a natural number, then please name a 1 in it, > > which cannot be covered, be it more at the front edge or more in the > > back area. Isn't this is a fair offer? > > No. Because I never argued such. You argued that every digit of K can be indexed. This means every digit can be covered. K is nothing else than every digit. Hence your claim unavoidably implies that K can be covered. This implies that K is in the true list. This is false. Again: Your claim that ***every*** digit of K can be indexed but K cannot be covered cannot hold unless you show a finite natural number which indexes a digit n but does not cover all digits less than n. Suppose I want to cover the n-th digit > of 0.111..., I simply take An. That does *not* show that 0.111... can > be covered by some An. Name 0.111... K. When K can be covered by any An > that would mean: > there is an n such that for all p in N, An[p] = K[p] (1) > the negation of that is (and that is what I argue): > there is no n such that for all p in N, An[p] = K[p] (2) > but that does *not* mean: > there is a p such that for all n in N, An[p] != K[p] (3) > which you seem to imply, by some garbled logic. My logic is very clear: (every digit of n can be indexed) ==> (n is in the list) not (n is in he list) ==> not (every digit of n can be indexed) Regards, WM |