Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Eckard Blumschein on 21 Nov 2006 10:23 For the first time I do not feel unnecessarily lectured and deliberately misunderstood. On 11/21/2006 3:07 PM, William Hughes wrote: > Eckard Blumschein wrote: >> While I do not question your last sentence being a correct conclusion if >> the premise holds, I cannot not see any possibility to reasonably and >> concisely quantify a largest possible number. >> >> Do we really have to define numbers? I tend to be satisfied when I have >> an executable rule which may create as many natural numbers as I like, >> as does Archimede's axiom. Executable means effectively about the same >> as communicable. It excludes postulated a priori existence. >> >> Do not get me wrong. I consider the properties executable and >> communicable like belonging to the abstract mathematical idea, not to >> any physical limitation to it. The process of counting indefinitely is >> never completely executable. Accordingly all created numbers are finite, >> and there is no principial reason for a largest possible number to be >> found. > > This is of course the standard position. It is not WM's position. I just described my position and wonder if it is the standard one. > On the larger question of whether the set of > all natural numbers exists. > > Your "executable rule" defines what is commonly referred to > as "potential infinity". Exactly. > The problem is the difference between saying > > The set of all natural numbers exists This is what I call Dedekind/Cantor Utopia. Elsewhere i tried to explain why it deals with sets instead of numbers. > and > We can generate an arbitrarily large set > of natural numbers Arbitrarily large is according to Cantor not really infinite. I share this view of him. > (i.e. the difference between actual and potential infinity) is > mostly one of terminology. Yes, but I got aware that actual and potential infinity are two different views both directed towards the same object of natural numbers from different levels of abstraction. The potential infinite view belongs to counting and is able to separate from each other all single numbers. The actual infinite one provides the fiction of all natural numbers at a higher level of abstraction. You cannot have both views at a time. > There is no element of the set > of natural numbers that does not exist as an element of some > arbitrarily large set. So there is absolutely no difference between > the > elements of "the infinite set of natural numbers" and the elements > of "the potentially infinite set of natural numbers". This is what I meant when writing above "the same object". > About the only thing you can say about the > "potentially infinite set of natural numbers" is that it is > not a set. So what? You still have this potential, > this thing that allows you to get new natural numbers > whenever you want. This potential has > all the properties of a set but the name. And the elusive belief hidden within and conveyed by this name. Aren't I correct? Eckard Blumschein > > - William Hughes >
From: Randy Poe on 21 Nov 2006 10:24 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > > > > > > > > > > > > The axiom of infinity says that the set N exists. Your iia says > > > > > > > > > > > > > > > > "we cannot recognize or treat all of its elements" > > > > > > > > > > > > > > This is no contradiction to the axiom. Compare the proof that the real > > > > > > > numbers can be well-ordered. We cannot construct or define or > > > > > > > recognize a well ordering. > > > > > > > > > > > > The fact that we cannot construct a real ordering does not > > > > > > stop us from proving such an ordering exists > > > > > > > > > > LOL. I know. > > > > > > > > > > > and using > > > > > > the existence of such an ordering. > > > > > > > > > > The existence of a well-order does not guarantee the constructibility > > > > > or definability of a real ordering. > > > > > The existence of a set does not guarantee the existence or definability > > > > > of a bijection or an identity mapping. > > > > > > > > No. > > > > > > > > To say that a set N exists is to say that all elements n of N exist. > > > > Thus the mapping n ->n for all elements n of N exists. > > > > > > That is exaggerated. There are models of ZFC (including countably many > > > elements a part of which could be interpreted as the set of natural > > > numbers) where no mapping on N exists. Why should it fail if you were > > > right? > > > > Piffle. Any defintion of mapping under which we can have > > an existing set of elements, but no existing identity map goes > > under the technical term of "stupid". > > That is a strong argument. > > You never heard of countable uncountable models? I've never heard of something which is both countable (can be bijected to N) and uncountable (can not be bijected to N). That would seem to require both P and (not P) to be true for some proposition P. > What about all contructible numbers or all words? They cannot be mapped > on N though they are a countable set. The set of all finite words, the set of polynomials, or any countable set can be put in bijection with N. Where do you get your view that it "can't be mapped to N"? - Randy
From: mueckenh on 21 Nov 2006 11:20 Ralf Bader schrieb: >Mückenheim gives ample proof of this > fact in his publications on the arxiv. In his paper "On a severe > inconsistency..." Mückenheims ascribes some assertions to Cantor, and > confirms this by quoting from Cantor's papers, which are obviously wrong. Please give an example. > If one checks this quotation, one sees that Mückenheim ignored assumptions > which Cantor had made previously in his text. Please give an example. > In another paper whose title > I don't remember It is the appendix to "On Cantor's important theorems: http://arxiv.org/ftp/math/papers/0306/0306200 > Mückenheim asserts to give a simpler proof of a theorem of > Cantor's, namely that the real plane stays connected if all points whose > coordinates are rational are left out. Comparing with Cantor's original, > one notices that Cantor had proved a much more general result, namely that > the real plane stays connected if an arbitrary countable set of points is > left out. The set of points with rational coordinates he later mentions as > an example, Later, that is after the 5th line: Was die abzählbaren Punktmengen betrifft, so bieten sie eine merkwürdige Erscheinung dar, welche ich im folgenden zum Ausdruck bringen möchte. Betrachten wir irgendeine Punktmenge (M), welche innerhalb eines n-dimensionalen stetig zusammenhängenden Gebietes A überalldicht verbreitet ist und die Eigenschaft der Abzählbarkeit besitzt, so daß die zu (M) gehörigen Punkte sich in der Reihenform ... vorstellen lassen; als Beispiel diene die Menge aller derjenigen Punkte unseres dreidimensionalen Raumes, deren Koordinaten in bezug auf ein orthogonales Koordinatensystem x, y, z alle drei algebraische Zahlenwerte haben. > and Mückenheim took that example for the main issue, ascribing > some other weird opinions to Cantor on the way. Although, surprisingly, > Mückenheims simplified proof of a restricted assertion is in principle > correct, it is also telling that he seems to consider it worth to be > published what actually is a mildly interesting exercise in basic point set > topology. Unfortunately you missed the clue. The simplified version has only been developed, because it can easily be extended to uncountable sets. I showed that the real plane even stays connected if an *uncountable* set of points is left out. So Cantor's theorem is useless. It doesn't show a difference between countable and uncountable, as Cantor might have tried to suggest. So, by your stupidity or your carelessness, you have again proven yourself a dishonest slanderer. However, thanks for your interest in my papers, even in the appendices, and for advertising them. Regards, WM
From: mueckenh on 21 Nov 2006 11:23 Dik T. Winter schrieb: > > I am discussing both language and maths. In German (as in Dutch) > there is a clear distinction between "Abzählen" and "Zählen". And as > you are in discussion with somebody from German origin, it is important > to keep this in mind. The first word means the process of counting, the > second means counting to get a result. In English for both the verb > "to count" is used (and is given as translation in dictionaries). I > think "to enumerate" is a better translation of "Abzählen". > > So let met rephrase the position of the opponents: > To enumerate the natural numbers you do not need [transfinite] ordinal numbers, > this is potential infinity. To count the natural numbers you do > need [transfinite] ordinal numbers. This is actual infinity. The first sentence is correct, but not exciting and not what Cantor meant. The second sentence is Cantor's position: Abzählen ins Unendliche. Regards, WM
From: mueckenh on 21 Nov 2006 11:24
Dik T. Winter schrieb: > In article <1163670499.235193.87050(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > If you want to find absolute truth you should not look at mathematics. > > > > Not at that what today is called mathematics, I agree. > > > > I + I = II is very fine and reliable mathematics. Absolutely true. And > > this approach can be put forward --- very far. > > What do you mean with those symbols? How do you define "+" and "="? > What is the meaning of "I" and "II"? What do you mean with "There" or "exists" or "a"? How do you define "set" and "which" and "has"? What is the meaning of "no" and "elements"? I think these symbols and expressions deserve closer examination than "+" and "II". Regards, WM |