Cantor Confusion
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From: Han de Bruijn on 8 Dec 2006 04:25 Virgil wrote: > In article <68588$45791ff3$82a1e228$8581(a)news2.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>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 > > A calculation is an application of mathematics, but may actually be > physics, or chemistry, or merely commerce. Geez! Can mathematics be separated from its "applications" in this way? Han de Bruijn
From: Eckard Blumschein on 8 Dec 2006 06:12 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. >> 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". Not even the tiniest interval including pi can be addressed point by point. You know, pi does not belong to the scale of countable numbers. Fictitious means uncountable. The reals must be fictitious in order to constitute the continuum. Countables are distinct from each other. > > 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. While Cantor's overcomprehensive sets are mysterious, a directly approachable numerical address is hopefully clearly understandable.
From: Eckard Blumschein on 8 Dec 2006 06:14 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.
From: Eckard Blumschein on 8 Dec 2006 06:15 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.
From: Bob Kolker on 8 Dec 2006 06:28
Eckard Blumschein wrote: > Knowing-alls could object: > 1) Pi has an exact numerical address (with an actually infinite amount > of let's say decimals). Define the concept "exact numerical address". Please do it plainly. Do you mean by "exactl numerical address" the decimal expansion or expansion in whatever base (greater than 1) is chosen? Bob Kolker |