Cantor Confusion
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From: stephen on 30 Nov 2006 20:26 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: >> > >> > In my tree there are no finite (terminating) paths! >> >> Sorry, as far as I see your tree has also finite paths. I see paths >> from the root to the first nodes below the roots. > That is an edge. But no path does stop there. Every path is continued, > infinitely. So you do not know what a path is, or a tree apparently. In a tree, there exists exactly one path between any two nodes. Stephen
From: Virgil on 30 Nov 2006 21:51 In article <eko0cd$ail$1(a)news.msu.edu>, stephen(a)nomail.com wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > >> > > >> > In my tree there are no finite (terminating) paths! > >> > >> Sorry, as far as I see your tree has also finite paths. I see paths > >> from the root to the first nodes below the roots. > > > That is an edge. But no path does stop there. Every path is continued, > > infinitely. > > So you do not know what a path is, or a tree apparently. > In a tree, there exists exactly one path between any two nodes. > > Stephen There are those for whom what connects *adjacent* nodes is called an edge, and others call it a branch. Most of those whom I am aware, other than possibly WM, reserve "path" for the chain of connections starting at the root node and ending, if at all, in a leaf node.
From: Eckard Blumschein on 1 Dec 2006 03:31 On 11/30/2006 9:44 AM, mueckenh(a)rz.fh-augsburg.de wrote: > Eckard Blumschein schrieb: > >> 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? > > Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: "Foundations > of Set Theory", 2nd edn., North Holland, Amsterdam (1984) Thank you. I only have the 1st ed. at hand. So the pertaining pages would perhaps not fit. Having had a very brief look into the book, I am surprized that the two authors (only Fraenkel and Bar-Hillel) confirm conclusions of mine. While Fraekel 1923 considered Brouwer one of the worst enemies of set theory, the late Fraenkel confirms that Brouwer and incidentally Heyting, too, just intended to save it differently. On the other hand, he did not hesitate to admit at p. 340 that "authors, among whom may be reckoned Poincar�, Brouwer, Wittgenstein, Kaufmann, Skolem, and Goldstein, arrive at their rejection of these transfinite operations from the observation that there exists no decision procedure for the truth of quantified statements. Identifying meaningfulness with effective veriffiability, they immediately arrive at the conclusion that sentences containing unlimited quantifiers are in general meaningless." Once again: _Sentences_ _containing_ _unlimited_ _quantifiers_ _are_ _in_ _general_ _meaningless_" Isn't this the reasonable and commonly agreeable on position of Galiei? Well Fraenkel does not likre this brutal truth. In 1923 he warned (p.164): "Wenn [der Angriff Brouwers] endgueltig glueckt, so bleibt abgesehen von engumgrenzten unangreifbaren Gebieten (namentlich der Arithmetik im engeren Sinn), von der gegenwaertigen Mathematik nur ein ungeheurer Truemmerhaufen uebrig." In 1958 he warns more modestly: "it would cripple mathematics just as the parallel view concerning empirical statements would cripple empirical sciences." Maybe serious mathematics has already been "crippled", and the cadaver of putative set theoretic foundation including all nonsensical notions like cardinality, transfinite numbers, alephs, betes, *R, and surreal numbers is just a holy monstrance for believers. I recall being a little boy wondering when I was told that while there is no evidence proving the existence of god there is also no evidence showing his non-existence. Are those crippled who don't believer in CH? I consider the background of CH given in the difference between number and continuum. This might be crippled down to the truth? Do you agree? >> >> > Then we have finite sets without >> > a largest element. >> >> Why not with a growing largest element? > > That depends on definition. The largest element of a set of numbers > today is not the same numbers as the largest element of the set > tomorrow. But the object "largest element" considered as a variable, in > fact would grow. > > Nevertheless this is not what Fraenkel et al. wish to express. They > talk about the development of the set of all sets in a Platonic world > view. I almost feel pity for Fraenkel who is challenged to disprove that "Cantor's ideas were but a pathological fancy" and has eventually to admit (p. 347) "the foundations of set theory are still somewhat shaky". Should we really try to retrace the fragile subtile lies of the authors? I don't think so.
From: mueckenh on 1 Dec 2006 05:09 William Hughes schrieb: > > 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. May be if you apply your personal definition of potentially infinity, but not if you apply the generally accepted definition. > > > > > > > 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. A potentially infinite set like N can be produced by induction. An actually infinite set cannot be produced by induction nor can it be produced by other means because it is simply nonsense. The only way of getting it is to postulate its existence by an axiom, although the degree of nonsense is not reduced by this method. > > 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. Try to learn the commonly accepted meaning of "potentially infinite". I am not wiling to discuss your personal definitions. > > 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. No. Your definition already fails with the meaning of "that can be shown to exist". In fact only the existence of a finite number of numbers can be shown to exist. So your set of real numbers is finite? Regards, WM
From: Eckard Blumschein on 1 Dec 2006 05:31
On 11/30/2006 1:38 PM, Bob Kolker wrote: > Eckard Blumschein wrote: > >> >> 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, > > Locke and Aristotle were NOT mathematicians. Aristotle was at most, a > logician or an ethicist or a political "scientist" or a literarary critic. > > As a scientist he was a failure. Why? He didn't check. > > Bob Kolker Weren't Aristotele, Galilei, Newton, Leibniz, Locke, Spinoza, Peirce, Poincar�, Einstein, Weyl so called universal scientists? Fermat who paved the way for calculus was not a mathematician but a lawyer. Borel was a politician. You are perhaps a mathematician. Do you believe to understand mathematics deeper than one out of the mentioned? Why should mathematics be esoteric? Eckard Blumschein |