Cantor Confusion
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From: Eckard Blumschein on 23 Nov 2006 07:49 On 11/23/2006 2:03 AM, William Hughes wrote: > Eckard Blumschein wrote: >> On 11/22/2006 1:46 PM, William Hughes wrote: >> > mueckenh(a)rz.fh-augsburg.de wrote: >> >> William Hughes schrieb: >> >> >> >> > No. You can develop "potential set theory". Just define >> >> > an element of a potential set to be an element of any one >> >> > of the arbitrary sets that can be produced. Now go through >> >> > set theory and add the word "potential" in front of each >> >> > occurence of the word set. There is no difference between >> >> > saying "a potentially infinite set exists" and saying "an actually >> >> > infinite set exists". >> >> >> >> A potentially infinite set has no cardinal number. It cannot be in >> >> bijection with another infinite set because it does not exist >> >> completely. >> >> Since I am convinced that Dedekind and Cantor were wrong, I avoid the >> expression cardinal number. > > Ok call the equivelence classes of the equivalence relation > bijection the breadfruits. They still exist. Already Galilei used bijection. Of course, bijection is feasable among genuine numbers without restrictions, no matter whether they constitute a finite set, the infinite series of natural numbers, rational numbers or the like. Therefore any mathematical object which is set together like a plurality of numbers is equivalent to the natural numbers. In other words: It is countable like peas in a cornet. Alternatively, powder in a cornet is equivalent to milk in a cornet in being uncountable. So we have two equivalence classes. I would like to abstain from the very risky preciptious claim that one of them is larger than the other. People who were less trained in abstraction may wonder why I reject the argument that powder consists of more particles than do the peas. My conterargument is: We consider powder like an ideal continuum. When I even reckon Dedekind and Cantor among those who adhere to the more intuitive argument claiming the existence of more powder than peas, I can quote indications for that on request. Let's summarize: We have two drawers, one mutually equivalent in being countable, the other one mutually equivalent in being uncountable. You suggested breadfruits, why not Alaaf? Citizen of Cologne used to shout Alaaf during carnival. Give them peas, and they have to shout Alaaf zero or simply Alaaf. Give them powder, and they have to shout Alaaf one. Have you any idea how to make them shouting Alaaf two? Three times Alaaf is possible: Alaaf! Alaaf! Alaaf! Just give them three times peas. However, what means Alaaf_two? Something can be false or correct. However, what does more than correct mean? >> Why do you not simply write instead: >> Something potentially infinite is countable because it is not thought to >> exhibit the impossible property to include all elements of something >> actually infinite? > > Because this is nonsense. Be cautious. > By defininition a set A is countable > if and only if there is a bijection between A and the natural numbers. Accepted. > Because of the way we Who? > have defined bijection and potentially > infinite set Cantor's definition of an infinite set has been withdrawn by Fraenkel without substitute because it was ambiguous with respect to the two mutually excluding points of view: actually and potentially. it does not matter at all whether or not A is actually > infinite when deciding if such a bijection exists. Proponents of set theory may hope that. >> If one considers something as actually infinite, then this point of view >> even hinders bijection. One cannot eat the cake and have it. >> potentially infinite and actually infinite point of view exclude each >> other as do discrete numbers and continuity. >> > > "One cannot eat the cake and have it" is not much of an > argument. It just illustrated the preceding sentence. > Do you have any other argument why you cannot > have bijections between potentially infinite sets. I did never deny bijection between natural numbers and any other plurality of genuine numbers. However, I came to the conclusion: There is no bijection between natural numbers, seen one by one, on one hand and the entity of all natural numbers, being understandable only like a fiction, on the other hand. In other words: While the realistic just potentially infinite set of IN is countable, the fictitious actually infinite set of IN is uncountable. Do not try to understand this from the perspective of set theory. Rather understand set theory a rather successful obscuration. Eckard Blumschein
From: Eckard Blumschein on 23 Nov 2006 08:05 On 11/22/2006 8:39 PM, Virgil wrote: > Something which is potentially, but not actually, > infinite cannot be a set at all in any set theory yet seen here. Isn't there consensus that the set IN of natural numbers is countable? How do you imagine bijection with a actually infinite set? Notice: Actual infinity is a "Gedankending" something fictitious. I do not say it is unconceivable or nonsense. It is just unapproachable. Moreove, potentially and actually infinite are mutually excluding points of view.
From: Franziska Neugebauer on 23 Nov 2006 08:23 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > That is not the question any longer. The question is: Can a set >> > theorist admit that she is in error? It seems impossible. They all >> > are too well trained in defending ZFC. >> >> In error as to what? Set theorists admit mistakes. It is not uncommon >> for books to have errata sheets attached. > > Would you see a contradiction in these two statements? > 1) "The cardinality of omega is |omega| not omega." > 2) "The cardinality of omega, also written as |omega|, is omega". As 2) may be considered as an "errata sheet" correcting the wording and rectifiying 1) there is no contradiction. Not in what I intended to say and not in some set theory anyhow. F. N. -- xyz
From: William Hughes on 23 Nov 2006 08:35 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > > If both, W and N, are countable, then renaming the elements of one of > > > > > them leads to an identity map. > > > > > > > > You are confusing bijections within the model to bijections > > > > outside of the model. > > > > > > And who told you that you were outside? > > And who told you that you were outside? > > > You are noting that it is not possible to construct a bijection > > in a nonstandard model and that it is > > possible to construct a bijection in the standard model. This > > is true, however, it is not a contradiction. > > In particular because there is neither a stadard model nor a > non-standard model of ZFC. In which case it is nonsensical to try to talk about Skolem's "paradox". You cannot have your cake and eat it too. - William Hughes
From: William Hughes on 23 Nov 2006 08:40
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > > > > > > The two statments > > > > > > > > i: P(n) is true for every n an element of N. > > > > ii: P(N) is true > > > > > > > > are not the same (trivial example P(x) is true iff x is an element of > > > > N). > > > > Induction can be used to prove statements of the form i. > > > > (eg all elements of N are finite). Induction cannot be used > > > > to prove statements of the form ii (e.g. N is finite). > > > > > > Here again your mathelogy comes to the surface. N is nothing but the > > > collection of all natural numbers. They count themselves. If all are > > > finite, then all are finite, > > > > Yes > > > > > i.e., then N is finite. > > > > No see above. The fact that all elements of N are finite > > does not mean that N is finite. > > It does. (The EIT proves it.) No it doesn't. Your claim is that assuming There exists an infinite set all of whose elements are finite leads to a contradiction. But you have yet to show a contradiction that doesn't require assuming There does not exist an infinite set all of whose elements are finite - William Hughes |