Cantor Confusion
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From: Eckard Blumschein on 4 Dec 2006 14:36 On 12/1/2006 3:10 PM, mueckenh(a)rz.fh-augsburg.de wrote: > 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 Why should we abandon the old and proven concepts number and continuum? To my understanding, they may or may not ideally fit the reality. Even after I know that solids consist of molecules, atoms, and smaller particles, there is no reason to start at this insight when designing let's say a building. Regards, Eckard
From: Virgil on 4 Dec 2006 14:39 In article <1165237792.210412.182250(a)80g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > We extend this to potentially infinite sets: > > > > A function from the set potentially infinite set A to the > > potentially infinite set B is a potentially infinite set of > > ordered pairs (a,b) such that a is an element of A and b is > > an element of B. > > > > We can now define bijections on potentially infinite sets > > and extend the bijection equivalence relation to include > > potentially infinite sets. Thus we can define > > equivalence classes under bijection of potentially infinite sets. > > Thus we can define "cardinal numbers" of potentially > > infinite sets. > > > There is only one "cardinal number". There is a different cardinal number for every different "size" of finite set, as well as for all these different "sizes" of non-finite sets. In order to apply any of Cantor's > proofs of higher cardinal numbers, a set of aleph_0 must be complete. > But it cannot be complete in potential infinity. So we need to get rid of potential infinity. Easy enough! Begone, thou pestilent idiocy! And gone it is.
From: stephen on 4 Dec 2006 14:39 imaginatorium(a)despammed.com wrote: > stephen(a)nomail.com wrote: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Dik T. Winter schrieb: >> >> >>Do they define sets as allowed to grow? Not in the quote you supply. >> >> There they talk about set valued variables that can grow. >> >> > No. X and Y do not grow, they remain "X" and "Y". The set they denote >> > does grow. The number of EC states may be n. "n" does not grow. The >> > number denoted by "n" does grow. >> >> What do you mean by the 'the number denoted by "n" does grow'? >> Currently the number of EC states is 25. In a month it will be 27. >> Does that mean 25 is going to grow into 27? Will 25 no longer exist? >> Or will 25 now mean 27? What do you mean by 'the number denoted by "25" does >> grow'? >> >> The idea that 25 is ever going to be anything but 25 is absolutely ridiculous. >> The idea that a set ever changes is equally ridiculous. > Obviously not a FORTRAN programmer... Not in quite some time. > SUBROUTINE CHANGE(A, B) > IF(A .EQ. 25) A = B > RETURN > END > Now try: > CALL CHANGE(25, 17) Are you claiming that FORTRAN is not ridiculous? :) Stephen
From: Virgil on 4 Dec 2006 14:47 In article <1165238765.397374.303270(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil 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. > > > > What "generally accepted" meaning is that? Most mathematicians do not > > accept that a set can be "potentially" infinite without being actually > > so. > > Most "mathematicians" even don't know what potentially infinite is. As it is a useless idea, such ignorance is bliss. And WM's sinful attempts to destroy that innocence is reprehensible. > > > > > > A potentially infinite set like N can be produced by induction. > > > > If it is a set, it is not merely "potentially" anything, and if it is > > merely potentially something then it is not a set. > > You do erroneously believe that you define the properties of a set. Sets in set theories with the axiom of extentionality must satisfy that axiom. And what does not satisfy it is not, in that system, a set. > > > > > > > An > > > actually infinite set cannot be produced by induction nor can it be > > > produced by other means because it is simply nonsense. > > > > That may be true in WM's as yet unspecified, and possibly unspecifyable, > > axiom system, but is false in ZF or NBG or NF. > > The impossibility of producing an actually infinite set by induction > should be obvious. In physical terms, sets have no existence at all. In the imagination, whatever can be imagined (and we can imagine infinite sets) exists in that imagination. > > > > > > > > > > > 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. > > > > The only way of getting anything in mathematics is by postulating it > > either directly or by postulating the means of constructing it. > > No. That is only necessary for non-existing entities ("existing" is > here to be understood as existence and not as belief). In the imagination, things only exist because we believe them to exist. It is only in the physical world that other criteria apply. > > > > > > > > Try to learn the commonly accepted meaning of "potentially infinite". > > > > There is no such thing as a 'commonly accepted meaning of "potentially > > infinite"'. > > > > > I > > > am not wiling to discuss your personal definitions. > > > > We do not wish to discuss your personal definitions either, but you keep > > thrusting them at us. > > Why don't you stop? Why don't you stop?
From: Eckard Blumschein on 4 Dec 2006 14:54
On 12/1/2006 3:03 PM, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > 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. The trick with these axioms is: They do not really define the notion set. Notice: Cantor's untennable definition has not been substituted by a new and correct one but the oracle-like axioms. The axiom of infinity has been stolen from solid ground of Archimedean potential infinity. You are correct: The wording "there is a set..." seems to imply the actually infinite point of view. However, isn't this merely cosmetics in order to nurture the illusion to cover the continuous zero-crossing? > Potential infinity is as impossible to describe as the set > of all set in ZF. The fabricators of set theory and its axiomatic crutch were perhaps much more intelligent than honest. Did you not realise that? > If ZF could use potential infinity, then this would be too honestly consequent. Z was unnecessary. Cardinality etc. were obviously futile. Real "numbers" were to be regarded mere fictions. Mathematics would not at all suffer but also not benefit much. |