Cantor Confusion
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From: Lester Zick on 8 Dec 2006 14:14 On Thu, 7 Dec 2006 19:38:44 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >stephen(a)nomail.com wrote: >> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> > stephen(a)nomail.com wrote: >> >> Nobody but you has talked about "growing" sets. Sets, like numbers, do not >> >> grow. You, like many other people who do not understand set theory, >> >> think of sets as mutable objects, that change as we perform operations >> >> on them. This is akin to thinking that numbers change when we perform >> >> addition. If I add 3 to 7, neither 3 or 7 changes. >> >> > It is such an odd belief. Why use a set for something that a function is >> > naturally for? I don't really understand why cranks insist on using sets >> > for everything, while at the same time insisting that sets are useless >> > or illogical or whatever. >> >> I do not think it is that odd. In everyday usage, the word "set" is used >> to denote something that changes. But then again, so is the word "number". >> The number of people in a room may change, but that does not imply that >> a specific number, such as 5, changes. Most people seem to have an abstract >> enough concept of number that the common usage does not confuse them. However >> they do not apply this abstraction to sets, so if someone says the set of people >> in the room changes, they think a specific set changes. > >That makes sense. And, most people have no understanding of the function >concept, so they don't see its utility or pervasiveness. Nor apparently its truth. ~v~~
From: Virgil on 8 Dec 2006 14:20 In article <45794891.9090207(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/8/2006 1:20 AM, David Marcus wrote: > > Mark Nudelman wrote: > >> On 12/7/2006 3:15 AM, Eckard Blumschein wrote: > >> > Just try and refute: > >> > > >> > Reals according to DA2 are fictitious > >> > Reals according to DA2 are fictitious > >> > With fictitious I meant: They must not have a directly approachable > >> > numerical address. This was the basis for the 2nd DA by Cantor afer an > >> > idea by Emil du Bois-Raymond. > >> > > >> > Good luck > >> > >> Can you define what you mean by a "directly approachable numerical > >> address"? > > > > He hasn't so far, despite being asked to. > > Knowing-alls could object: > 1) Pi has an exact numerical address (with an actually infinite amount > of let's say decimals). > 2) sqrt(2) has an exact numerical address (goto 2 and then just > calculate the root). > 3) 1.0000000000000000... (with an actually infinite amount of decimals) > is just the same as 1. > > > > > >> Do you mean that SOME reals don't have such an address, or that NONE of > >> them have such an address? > > None of them. > > >> Is "sqrt(2)" a "directly approachable > >> numerical address"? > > No. > > >> If you're saying that some reals do not have a finite decimal > >> representation, that's true, > > No. All, not some, reals (as indirectly defined by DA2) do not have a f.d.r. Wrong! All reals have a non-ending decimal representation, but that does not preclude some of them from also having a finite one. For some reals, their decimal representations have non-zero digits in only finitely many digit positions, and all such also have finite decimal representations. > > >> but so what? Why does that make them any > >> more "fictitious" than the integers? > > I do not use the word fictitious for integers. The lacking address > corresponds to the impossibility to count these "numbers". One can "count" all sorts of sets of your pseudo-fictitious numbers, for example the set of square roots of primes is as countable as the set of primes. > Not even the > tiniest interval including pi can be addressed point by point. The set of rationals in such an interval can also not be addressed point by point. You know, > pi does not belong to the scale of countable numbers. Fictitious means > uncountable. Then say "uncountable" where you mean uncountable. > The reals must be fictitious in order to constitute the > continuum. You mean that they must be uncountable! Countables are distinct from each other. The countable rationals are dense. Between any two of them there are infinitely many others. How are they any more "distinct from each other" that reals? > > > > > Ah, but it sounds better to say they don't have a "directly approachable > > numerical address" than to say they don't have a "finite decimal > > representation". Much more mysterious-sounding. > > Decimal address is only one option. Other that using a non-decimal base, what other options are there? > > While Cantor's overcomprehensive sets are mysterious, a directly > approachable numerical address is hopefully clearly understandable. Sqrt(2) has a directly approachable decimal numerical address.
From: cbrown on 8 Dec 2006 14:21 Han de Bruijn wrote: > Lester Zick wrote: > > > On Thu, 7 Dec 2006 03:00:21 -0500, David Marcus > > <DavidMarcus(a)alumdotmit.edu> wrote: > > > >>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. > > > > So you're saying that set "theory" is all of mathematics? Of course > > since what you say isn't necessarily true that's not exactly a ringing > > endorsement of set "theory". > > It's quite simple. Set Theory can not be the foundation for mathematics, > because NOT EVERYTHING IS A SET. E.g. a calculation is mathematics, but > it's not a set. But calculations performed on the real numbers can, in principle, be translated to operations on sets and back again to obtain identical results. Of course, we might disagree whether something is mathematics, just as we might disagree over whether something is an example of Italian cooking. Can you give an example of something you consider mathematical that you feel cannot be modeled using sets as described by ZFC? > Set theory may be of limited use, but it's supremacy is > complete nonsense, and will be overruled in time. "Supremacy"? It's not somehow "better" than other branches of mathematics. The fact that in principle we can perform calculations equivalently by performing certain set theoretic operations doesn't mean that the latter is a "good" way of performing calculations. Set theory is primarily a tool used to unify different branches of mathematics by providing these different branches with a common language. Cheers - Chas
From: Virgil on 8 Dec 2006 14:23 In article <4579492B.1000206(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/8/2006 1:15 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> 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 > > > > Oh, well, that clears everything up. > > Read completely what I wrote and be ashamed of your arrogance. I have read it and find the arrogance is all EB's. As well as misrepresentations and errors.
From: Virgil on 8 Dec 2006 14:26
In article <45794968.4020405(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/8/2006 1:09 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> On 12/7/2006 4:38 AM, Virgil wrote: > >> > In article <4576DF19.7070005(a)et.uni-magdeburg.de>, > >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> > >> >> The abstract concept of numbers > >> >> must not be misused as to declare rationals and embeded rationals > >> >> likewise existent. > >> > > >> > The "abstract concept of number" can be used in any way that > >> > mathematicians choose to use it, > >> > >> If there was really general agreement among mathematicians, then there > >> would be an acceptable printed definition. Since Cantor's definition of > >> set has been declared untennable without substitute, I do not expect a > >> clean definition of number either. > > > > What definition of set is this and why is it untenable? > > > > I replied today tomorrow. Looks like "manana" to me. |