Cantor Confusion
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From: Eckard Blumschein on 30 Nov 2006 04:45 On 11/29/2006 7:41 PM, MoeBlee wrote: > Eckard Blumschein wrote: >> On 11/29/2006 5:24 PM, mueckenh(a)rz.fh-augsburg.de wrote: >> We have to accept thart there are sets which are capable of >> > growing, as Fraenkel et al. expess it. >> >> While there are several books by Fraenkel: >> >> Einleitung in die Mengenlehre 2nd ed. 1923, 3rd ed. 1946, >> Gesammelte Abhandlungen Cantor, Dedekind 1932 >> Das Leben Georg Cantors 1932 >> Abstract set theory 1961 >> Lebenskreise - Aus den Erinnerungen eines j�dischen Mathematikers 1967 >> >> perhaps there is only one book by Fraenkel et al.: >> Foundations of Set Theory 1958. Correct? >> >> > Then we have finite sets without >> > a largest element. >> >> Why not with a growing largest element? > > Because Fraenkel, Bar-Hillel, and Levy said nothing that implies any > such thing as that there are finite sets without a largest element. WM > is just latching onto a particular quote(that even uses scare quotes) > out of context and with utter disregard to what the quote is meant to > actually summarize. I will look into the book myself. Everyone of us my be wrong in details. I guess, we cannot be cautious enough with respect to possibly changing meaning of words.
From: mueckenh on 30 Nov 2006 04:45 Virgil schrieb: > The set of transcendentals is not a connected set in R, so violates the > hypotheses of the proof. So the set of transcendentals cannot be proved uncountable by this proof. Nor can it be proved uncountable by the diagonal argument. Do you have an idea why this is so? Could it be that Cantor's proof have not at all to do with countability? In my paper I stress the fact that his proof is equally vaild or invalid, when applied to the rational numbers alone as well as when applied to the transcenental numbers alone. What is wrong with this statement? Regards, WM
From: Eckard Blumschein on 30 Nov 2006 05:09 On 11/29/2006 7:41 PM, Virgil wrote: > In mathematics, claims unsupported by mathematically > valid proofs do not persuade, particularly when opposed by > mathematically valid proofs of their falsehood. Really? Fraenkel, 2nd ed. 1923, approved Cantor having managed not just to battle but also refute an assertion by Gauss. Is there any evidence for this proud claim? No. Whenever Cantor declared mathematicians like Aristotele, Locke, Descarte, or Leibniz proven wrong, he did it with evidence by assertion. His main argument DA2 convinced a lot of mathematicians because understanding of infinity requires more that mathematical education, and the attention focussed on the flawless evidence for non-countablilty. So Cantors invalid interpretation of DA2 persuaded. Dedekind admitted at the very beginning that he did not have a proof.
From: Eckard Blumschein on 30 Nov 2006 05:20 On 11/29/2006 7:47 PM, Virgil wrote: > In article <456DC03A.1000206(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/29/2006 5:24 PM, mueckenh(a)rz.fh-augsburg.de wrote: >> We have to accept thart there are sets which are capable of >> > growing, as Fraenkel et al. expess it. >> >> While there are several books by Fraenkel: >> >> Einleitung in die Mengenlehre 2nd ed. 1923, 3rd ed. 1946, >> Gesammelte Abhandlungen Cantor, Dedekind 1932 >> Das Leben Georg Cantors 1932 >> Abstract set theory 1961 >> Lebenskreise - Aus den Erinnerungen eines j�dischen Mathematikers 1967 >> >> perhaps there is only one book by Fraenkel et al.: >> Foundations of Set Theory 1958. Correct? >> >> > Then we have finite sets without >> > a largest element. >> >> Why not with a growing largest element? >> >> Regards, >> Eckard > > Growing elements would allow claims like: > "2 + 2 = 5 for large enough values of 2". Large enough is certainly not qualitatively different enough, infinity is the location where two parallel lines are thought to meet each other, and division by zero has been forbidden because it yields anything. Since I consider an infinite set just like a possibly useful fiction, I do not have any reason to speculate myself. I am just curious what monstrosity of split reasoning in best Cantorian tradition Fraenel et al. will offer.
From: William Hughes on 30 Nov 2006 07:22
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > So what we have to do to prove that all sets of natural numbers > > > > are finite is to assume that there is no set of natural numbers > > > > without a largest element. > > > > > > No. We have to accept thart there are sets which are capable of > > > growing, as Fraenkel et al. expess it. Then we have finite sets without > > > a largest element. Then we describe reality correctly. > > > > (Leave Fraenkel et al. out of it. Finite sets without a largest > > element are your idea). > > I did not ascribe those sets to Fraenkel et al. I said, "growing, as > Fraenkel et al. express it." And, in fact, they used this expression. > > > > O.K. Let A be a finite set without a largest element. > > What is the cardinal number of A. Clearly it cannot > > be any finite cardinal number? > > It cannot have a cardinal number at all. We have no cardinals in > potential infinity. Cantor knew that. Piffle. Extending the concept of bijection from sets to potentially infinite sets is trivial. > > > > Recall, you want to do more than just state that A does > > not have a cardinal number. You want to show that assuming > > that A has a cardinal number leads to a contradiction. The problem is although the term "finite" is used in the definition of A, A does not have many of the properties that are usually associated with finite sets. In particular it is not possible to produce A by induction. It is clear that there is no natural number n, such that there is a bijection between the set {1,2,3....,n} and A. Since the diagonal of your matrix has no largest element, it must be a "finite set without largest element". It is clear that there is no bijection between one line and the diagonal. > Please do not mix up potential and actual infinity. Actual infinity is > required, e.g., for the real numbers. Piffle. The rational numbers form a potentially infinite set A. Let B and C be two potentially infinite sets such that: For any element a, that can be shown to exist in A, it is possible to show that a exists in exactly one of B and C. for any element, b, that can be shown to exist in B, for any element, c, that can be shown to exist in C and b < c. for any element b_1 that can be shown to exist in B, there is an element b_2 that can be shown to exist in B, and b_1 < b_2 The potentially infinite set of pairs (B,C) is the real numbers. - William Hughes |