Cantor Confusion
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From: William Hughes on 1 Dec 2006 08:40 mueckenh(a)rz.fh-augsburg.de wrote: > 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. Informally we have that a potentially infinite set is a set which is always finite, but to which we can add an element whenever we want. We say that x is an element of the potentially infinite set if we can add enough elements to get to x. More formally. Let a generalized set be a function on sets which takes the value true or false. Let A be a generalized set. We say that a set B is a subset of the generalized set if A(B) is true. We say that x is an element of the generalized set if there exists a set C such that A(C) is true and x is an element of C. A generalized set D, is potentially infinite if: for any set E such that D(E) is true, there exists a set F such that E is a proper subset of F and D(F) is true. Do you have a different definition of potentially infinite set? > > > > > > > > > > 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. Every set "produced by induction" has a largest element. The set N does not have a largest element. >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. How does the definition I am using differ from the 'commonly accepted meaning of "potentially infinite"'? > > > > 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? The set of numbers is potentially infinite. So the real numbers are potentially infinite. - William Hughes
From: mueckenh on 1 Dec 2006 09:03 William Hughes schrieb: > > > 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. > > > Informally we have that a potentially infinite set is a set > which is always finite, but to which we can add an element > whenever we want. We say that x is an element of > the potentially infinite set if we can add enough elements > to get to x. Exactly. Therefore the potentially infinite sequence of digits of a real number is always representing a rational number, even a rational number with a finite sequence of digits. > > More formally. > > Let a generalized set be a function > on sets which takes the value true or false. Let A be > a generalized set. We say that a set B is a subset > of the generalized set if A(B) is true. We say that x > is an element of the generalized set if there exists > a set C such that A(C) is true and x is an element of C. > A generalized set D, is potentially infinite if: for any > set E such that D(E) is true, there exists a set F such > that E is a proper subset of F and D(F) is true. > > Do you have a different definition of potentially > infinite set? If you use the term "set" (like for instance for your set A) as defined in set theory, then all the elements are "there" (where ever that may be). Therefore you cannot describe potential infinity by means of ZF or NBG set theory, unless you use completely different definitions of "set" etc. Potential infinity is as impossible to describe as the set of all set in ZF. If ZF could use potential infinity, then the set of all sets would not be a problem, because this set which obviously exists, is potentially infinite. > > > > > > > > > > > > > > > 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. > > Every set "produced by induction" has a largest element. > The set N does not have a largest element. > A set produced by induction has a largest element, but the set and with it the largest element can change. > > > > > > 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. > > > How does the definition I am using differ from the > 'commonly accepted meaning of "potentially infinite"'? If you understand German, here are two of the best definitions I ever saw: Will man in Kürze die neue Auffassung des Unendlichen, der Cantor Eingang verschafft hat, charakterisieren, so könnte man wohl sagen: in der Analysis haben wir es nur mit dem Unendlichkleinen und dem Unendlichgroßen als Limesbegriff, als etwas Werdendem, Entstehendem, Erzeugtem, d.h., wie man sagt, mit dem potentiell Unendlichen zu tun. Aber das eigentlich Unendliche selbst ist dies nicht. Dieses haben wir z.B., wenn wir die Gesamtheit der Zahlen 1,2,3,4, ... selbst als eine fertige Einheit betrachten oder die Punkte einer Strecke als eine Gesamtheit von Dingen ansehen, die fertig vorliegt. Diese Art des Unendlichen wird als aktual unendlich bezeichnet. [David Hilbert: Über das Unendliche, Math. Ann. 95 (1925) p. 167] Daß das sogenannte potentiale oder synkategorematische Unendliche (Indefinitum) zu keiner derartigen Einteilung Veranlassung gibt, hat darin seinen Grund, daß es ausschließlich als Beziehungsbegriff, als Hilfsvorstellung unseres Denkens Bedeutung hat, für sich aber keine Idee bezeichnet; in jener Rolle hat es allerdings durch die von Leibniz und Newton erfundene Differential- und Integralrechnung seinen großen Wert als Erkenntnismittel und Instrument unseres Geistes bewiesen; eine weitergehende Bedeutung kann dasselbe nicht für sich in Anspruch nehmen. [G. Cantor, Gesammelte Anhandlungen, p. 373] Trotz wesentlicher Verschiedenheit der Begriffe des potentialen und aktualen Unendlichen, indem ersteres eine veränderliche endliche, über alle Grenzen hinaus wachsende Größe, letztere ein in sich festes, konstantes, jedoch jenseits aller endlichen Größen liegendes Quantum bedeutet, tritt doch leider nur zu oft der Fall ein, daß das eine mit dem andern verwechselt wird. [G. Cantor, Gesammelte Anhandlungen, p. 374] In short: A potentially infinite entity is a variable, always finite but surpassing every border. The atually infinite is fixed, i.e., it is a quantity. > The set of numbers is potentially infinite. So the real numbers > are potentially infinite. That is my opinion. But it was my impression that you would rather believe in a fixed quantity. Regards, WM
From: mueckenh on 1 Dec 2006 09:10 Eckard Blumschein schrieb: > 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? > No, I am sorry, I do not. The continuum is nothing but our failure to look closely enough. In physics it lasted 2000 years to settle the idea of the atom and to supplement and complete it by the uncertainty relations. The majority of matematicians is not yet far sighted enough to recognize the same situation in their realm. Regards, WM
From: William Hughes on 1 Dec 2006 09:29 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > 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. > > > > > > Informally we have that a potentially infinite set is a set > > which is always finite, but to which we can add an element > > whenever we want. We say that x is an element of > > the potentially infinite set if we can add enough elements > > to get to x. > > Exactly. Therefore the potentially infinite sequence of digits of a > real number is always representing a rational number, even a rational > number with a finite sequence of digits. > > > > More formally. > > > > Let a generalized set be a function > > on sets which takes the value true or false. Let A be > > a generalized set. We say that a set B is a subset > > of the generalized set if A(B) is true. We say that x > > is an element of the generalized set if there exists > > a set C such that A(C) is true and x is an element of C. > > A generalized set D, is potentially infinite if: for any > > set E such that D(E) is true, there exists a set F such > > that E is a proper subset of F and D(F) is true. > > > > Do you have a different definition of potentially > > infinite set? > > If you use the term "set" (like for instance for your set A) as defined > in set theory, then all the elements are "there" (where ever that may > be). Therefore you cannot describe potential infinity by means of ZF or > NBG set theory, unless you use completely different definitions of > "set" etc. Potential infinity is as impossible to describe as the set > of all set in ZF. If ZF could use potential infinity, then the set of > all sets would not be a problem, because this set which obviously > exists, is potentially infinite. O.K. we now know that there is no concept of potential infinity in ZF (this comes under the heading "the pacific ocean is wet"). However, you have not actually given a defintion of potential infinity. Do you have one? > > > > > > > > > > > > > > > > > > > > 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. > > > > Every set "produced by induction" has a largest element. > > The set N does not have a largest element. > > > A set produced by induction has a largest element, but the set and with > it the largest element can change. > > > > > > > > > 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. > > > > > > How does the definition I am using differ from the > > 'commonly accepted meaning of "potentially infinite"'? > > If you understand German, here are two of the best definitions I ever > saw: > > Will man in Kürze die neue Auffassung des Unendlichen, der Cantor > Eingang verschafft hat, charakterisieren, so könnte man wohl sagen: in > der Analysis haben wir es nur mit dem Unendlichkleinen und dem > Unendlichgroßen als Limesbegriff, als etwas Werdendem, Entstehendem, > Erzeugtem, d.h., wie man sagt, mit dem potentiell Unendlichen zu tun. > Aber das eigentlich Unendliche selbst ist dies nicht. Dieses haben wir > z.B., wenn wir die Gesamtheit der Zahlen 1,2,3,4, ... selbst als eine > fertige Einheit betrachten oder die Punkte einer Strecke als eine > Gesamtheit von Dingen ansehen, die fertig vorliegt. Diese Art des > Unendlichen wird als aktual unendlich bezeichnet. [David Hilbert: Über > das Unendliche, Math. Ann. 95 (1925) p. 167] > > Daß das sogenannte potentiale oder synkategorematische Unendliche > (Indefinitum) zu keiner derartigen Einteilung Veranlassung gibt, hat > darin seinen Grund, daß es ausschließlich als Beziehungsbegriff, als > Hilfsvorstellung unseres Denkens Bedeutung hat, für sich aber keine > Idee bezeichnet; in jener Rolle hat es allerdings durch die von Leibniz > und Newton erfundene Differential- und Integralrechnung seinen großen > Wert als Erkenntnismittel und Instrument unseres Geistes bewiesen; eine > weitergehende Bedeutung kann dasselbe nicht für sich in Anspruch > nehmen. [G. Cantor, Gesammelte Anhandlungen, p. 373] > > Trotz wesentlicher Verschiedenheit der Begriffe des potentialen und > aktualen Unendlichen, indem ersteres eine veränderliche endliche, > über alle Grenzen hinaus wachsende Größe, letztere ein in sich > festes, konstantes, jedoch jenseits aller endlichen Größen liegendes > Quantum bedeutet, tritt doch leider nur zu oft der Fall ein, daß das > eine mit dem andern verwechselt wird. [G. Cantor, Gesammelte > Anhandlungen, p. 374] > > In short: A potentially infinite entity is a variable, always finite > but surpassing every border. The atually infinite is fixed, i.e., it is > a quantity. > Nothing in the above, in either language, precludes in any way giving a meaning to "x is an element of the potentially infinite entity A". - William Hughes
From: mueckenh on 1 Dec 2006 09:42
William Hughes schrieb: > O.K. we now know that there is no concept of potential infinity > in ZF (this comes under the heading "the pacific ocean is wet"). > However, you have not actually given a defintion of potential > infinity. Do you have one? Take the Peano axioms, in particular the axiom of induction. They show a precise definition of what a potentially infinite set is. > > > > > >als etwas Werdendem, Entstehendem, > > Erzeugtem, d.h., wie man sagt, mit dem potentiell Unendlichen zu tun. [David Hilbert: Über > > das Unendliche, Math. Ann. 95 (1925) p. 167] > > In short: A potentially infinite entity is a variable, always finite > > but surpassing every border. The atually infinite is fixed, i.e., it is > > a quantity. > > > > Nothing in the above, in either language, precludes in > any way giving a meaning to > "x is an element of the potentially infinite entity A". That is correct. 7 is an element of the set of natural numbers. But this set is not a set in the sense of set theory. I.e., it has no cardinal number which is a quantity. It has at most the "cardinal number" oo. But that is a property, not a quantity. Regards, WM |