Cantor Confusion
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From: David Marcus on 7 Dec 2006 02:46 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > And the power set of this set is a finite set too. And so on, in > > > infinity ... (potential infinity , of course) > > > > You still have not yet understood the concept of inifinite sets. > > I have understood the concept and its failure. If you've really understood it, please demonstrate this. State a theorem and give its proof in standard set theory. -- David Marcus
From: David Marcus on 7 Dec 2006 02:54 Virgil wrote: > In article <1165421742.266029.197420(a)79g2000cws.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Bob Kolker schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > I have understood the concept and its failure. (That is the parallel > > > > between those who have not yet arrived and those who have already left: > > > > Both are not there.) > > > > > > What failure? > > > > One of many examples: The set {2,4,6,...,2n} has a cardinal number less > > than some numbers in the set. This does not change when n grows (yes, > > it can grow!) over all upper bounds. Therefore the assertion that the > > set of all even natural numbers has a cardinal number gretaer than any > > even number is false. > > It appears that the negation of "the set of all even numbers has > cardinal greater than any even number" would have to be " there is an > even natural number as great as the cardinality of the set of all even > natural numbers." > > If I am misunderstanding, WM, let him state his own understanding of > what that negations wold be, but without merely appending "is false". WM is just repeating his incorrect use of limits. Start with |{2,4,...,2n}| < 2n. Letting n go to infinity, WM gets |{2,4,...}| < w. Or, something like that. It is hard for me to keep track of all of his wrong arguments. He really is very inventive in coming up with fallacies. -- David Marcus
From: David Marcus on 7 Dec 2006 03:00 Bob Kolker wrote: > MoeBlee wrote: > > In the sense you're trying to get across to the other poster, I > > understand your point. But, just for the record, in a technical sense > > in set theory, as integers, rational numbers, and real numbers are > > themselves sets, it does make sense to say whether one of them is > > countable or not. For example, where integers are defined as > > equivalence classes of natural numbers, each integer is itself a > > denumerable set. I am not necessarily endorsing anything the other > > poster has said; I'm just adding the technical note that in a strict > > set theoretic sense, even numbers are sets and thus it is meaningful to > > talk about the cardinality of a number. > > an element of a ring or a semi-group is a set? If you use ZFC (or something similar) as your foundation for mathematics, then everything is a set. Of course, while solid foundations are good to have, if you are living on an upper floor, you may prefer to ignore what is going on in the basement. -- David Marcus
From: mueckenh on 7 Dec 2006 05:36 Franziska Neugebauer schrieb: > >> This does not imply that the set of former times "has changed" in > >> time. Only the naming has changed due to a changed definition. You > >> may compare this with Gerhard Schröder who did not undergo a gender > >> transformation when Angela Merkel became chancellor in 2005. > >> > > But "the chancellor" did. > > I would think most people start laughing at the questioner when asked > whether > > In 2005 the chancellor underwent a gender transformation. > > is true. Many people start laughing in case they fail to understand. Only few start thinking. Regards, WM
From: Eckard Blumschein on 7 Dec 2006 05:45
On 12/7/2006 5:26 AM, Virgil wrote: > In article <4576FE21.8040606(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/6/2006 2:05 AM, Virgil wrote: >> > In article <1165345832.735910.255620(a)79g2000cws.googlegroups.com>, >> > "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> > >> >> Bob Kolker wrote: >> >> > There is no such thing as a countable integer, countable rational or >> >> > countable real. >> >> >> >> In the sense you're trying to get across to the other poster, I >> >> understand your point. But, just for the record, in a technical sense >> >> in set theory, as integers, rational numbers, and real numbers are >> >> themselves sets, it does make sense to say whether one of them is >> >> countable or not. For example, where integers are defined as >> >> equivalence classes of natural numbers, each integer is itself a >> >> denumerable set. I am not necessarily endorsing anything the other >> >> poster has said; I'm just adding the technical note that in a strict >> >> set theoretic sense, even numbers are sets and thus it is meaningful to >> >> talk about the cardinality of a number. >> >> >> >> MoeBlee >> > >> > In that particular sense, each real number as a Dedekind cut, being a >> > partition of the rationals into two sets, always has cardinality 2, >> > while each member of a Dedekind cut is countably infinite. >> > >> > But each real number as a set of equivalent Cauchy sequences is >> > uncountable. >> >> Conclusion: Dedekind cut's are self-delusive. > > Why is any set with two clearly defined members self-delusive? Virgil, While I understood you refer to subsets, I would like to explain the whole delusion first. The "number" pi is definitely a merely fictitious element of continuum. It is clearly defined by a geometrical problem which cannot be solved numerically by means of a realistic, i.e. finite number of steps. There is no possibility to reasonably quantify the amount of such fictitious elements. The continuum of such "elements" is uncountable, no more and no less than anything which is considered perfectly infinite. Notice: Actual infinity means to abstractly include _all_ of indefinitely many naturals, integers, rationals, irrationals, or reals. When I wrote "abstractly", I meant it is impossible to reach infinity with counting. Archimedes quasi defined natural numbers like someting that can indefinitely be enlarged by just adding one more unit. Likewise fractional numbers can be indefinitely reduced. So rational numbers represent the Archimedean and Aristotelean notion of the potentially indefinitely large and also the indefinitely small. Because the term Archimedean has been given a deviating definition, I call such numbers genuine numbers. You may argue: The expression rational numbers is sufficient. Well, you are correct. I intend to stress that only rational numbers including intergers and naturals are genuine. Moreover, rational numbers loose their property of being countable if they are embedded into the continuum. It would not be wrong to interprete this loss of the property to be countable as loss of existence. At least there is no possiblity to decide inside the genuine continuum whether a fictitious "element" belongs to the rationals or to the irrationals except via the defining problem in each case. The primary continuum is strictly speaking amorph. There is no structure available inside this continuum. Alleged homomorphy is valid for rational quasi-reals. Ascribing the behavior of genuine numbers to the reals is tempting but not justified. Already Cauchy did not care about the categorical distinction between rationals and reals. E. Heine "Die Elemente der Funktionenlehre", Crelles Journal, Bd. 74 further encouraged to do so. I guess, there is indeed no compelling reason to strictly obey the correct categories in practice. What illusions I refer to? 1) Dedekind dreamed of making the rationals complete by addition of numbers in between two rationals. This is neither possible nor necessary because already systems of rational numbers are everywhere dense. It is impossible to make reals rational, to make infinity a finite quantum, and to resolve the continuum into countable points. 2) Dedekind imagined a line composed of single points. He argued: These points are continuously ordered form left(small) to right (large). He ignored that these points are just fictitious ones even if they correspond to the solution of a geometrical problem. He was still correct when he wrote that every rational number corresponds to only one single point. Was he still correct in that there are indefinitely many points which do not correspond to a rational number? Seemingly yes. However, his idea that there are more reals than rationals tacitly presumes: The entities of all rationals and all reals within a common interval can be quantified and ergo can be compared with each other. 3) Dedekind as well as the majority of mathematicians believed to be entitled to decide this question intuitively. It seems to be quite clear to them that there are much more rational numbers than real ones because the rational numbers are included within the reals. Consequently the number or reals must be larger than indefinitely large. 4) Dedekind wrote: "Zerfallen alle Punkte der Geraden in zwei Klassen von der Art, dass jeder Punkt der ersten Klasse links von jedem Punkt der zweiten Klasse liegt, so existiert ein und nur ein Punkt, welcher diese Einteilung aller Punkte ... hervorbringt". In brief: D. assumed the line to consist of "all" points, and these points have to be located either left or right with respect to just one selected point. He admitted to be unable to prove this. Indeed this idea was wrong if we allow for indefinitely many points. In order to select a point, we have to have all points first. This is impossible. 5) Dedekind claimed to be in position to create real numbers by means of his cuts, obviously with no avail. In order to know whether or not a number is irreal, one has to define it first. 6) Admittedly up to now, I myself I was taken in by Dedekind's elusive intuition. As did Stifel and Weyl, I correctly imagined the entity of all real numbers continuous like a fog or a sauce while I imagined the rationals as ordered single points. Wrong was just the expression "the" rationals. Any set of rational numbers corresponds to insulated points being different from each other. "The" means all. However, all rationals are a fiction, the same foglike fiction as are the reals. So the difference between rational and real numbers is actually merely a categorical one. In other words, it depends on the point of view. Take the position of counting: Genuine numbers are considered countable even if they are as dense as a fog. Take the opposite position: The genuine continuum is considered to consist of uncountable reals while approximated by dots is sufficient in practice. 7) I checked whether or not the difference between rationals and irrationals is indeed merely a categorical one: If irrationality has been proven by showing that a common divisor is missing, then this is bound to quantities of finite size. Example 2/2=1 but 2000000000...000000001/2000000000...000000000 =/= 1 In other words: I cannot confirm the difference between rationals and reals, closed and open intervals, countable and uncountable, digital and analog, etc. to persist where the realms of genuine numbers and the genuine continuum are thought to meet each other. Dr.-Ing. Eckard Blumschein Electrical engineer, Uni of Magdeburg |