Cantor Confusion
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From: Eckard Blumschein on 23 Nov 2006 02:31 On 11/23/2006 6:37 AM, Virgil wrote: > Eckard Blumschein wrote: >> "Why at all do we consider sets instead of numbers?". My feeling says >> apriorism is a special kind of stupidity and a very stubborn one. > > > Those whose "feelings" tell them that they are Napolean Buonaparte have > feelings as relevant to the structure of mathematics as EB's. Admittedly, the word apriorism does not always mean dogmatism. Look into httr://wiki.cotch.net/index.php/Apriorism as to find: Apriorism: Explanation: You commit this fallacy if you reason from abstract principles to facts not vice versa. Demarcation: You do not commit this fallacy if you draw conclusions from abstract principles, unless you call these conclusions facts. "Attributing concreteness to the abstract" belongs to the reification fallacy. For further reading look into http://www.philosophyprofessor.com/philosophies/apriorism.php So I have to describe more precisely what I meant with apriorism.
From: Eckard Blumschein on 23 Nov 2006 03:02 On 11/23/2006 6:58 AM, Virgil wrote: > In article <45646A1E.1080606(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> Virgil wrote: >> > >> > Eckard Blumschein wrote: >> > >> >> William Hughes wrote: >> >> >> >> > As you stated "Goes on forever" is a property of the natural >> >> > numbers. The set {1,2,3,...} is just the natural numbers. So >> >> > this set must go on forever. >> >> Eckard>> >> Being admittedly not very familiar with set theory, I nonetheless wonder >> >> if sets are considered like something going on forever. >> > William>> > Sequences do, sets do not. >> > Virgil>> > When one represents the members of a set as members of a sequence, as >> > {1,2,3,...} does, that little ambiguity should not really mislead anyone. >> Eckard>> That little ambiguity? Doesn't it make a categorical difference if a set >> has been a priori set for good? > Virgil> When one speaks of the sequence {1,2,3,...} and another speaks of the > set {1,2,3,...}, the notational ambiguity should not override the words > "sequence" and "set". Perhaps you did not get or at least did not accept my point: Does it not make a categorical difference whether a set is something unchanging beeing already perfect for good or something that just formally includes the vital property of the series of nagtural numbers to have no end at all? The notation {1, 2, 3, ...} is ambiguous with this respect. It depends on the decision whether ... denote potential or actual infinity. Meanwhile I see mounting evidence for my suspicion that set theory is some sort of (self?)-deception. It claims to rule continuity and incorporate the irrational ratios. However axiomatic set theory merely replaces Weierstrass's way to formalize Cauchy's idea. I do not deny that this is clever. However, the whole burden of Dedekind's cuts and Cantor's indeed naive fabrication of cardinality, cardinal and ordinal numbers, transfinite numbers and more than infinitely many numbers does not have any sound basis. Real numbers are defined in a misleading manner. Therefore present mathematical education hinders to understand my reasoning that the genuine continuum IR_g _can_ be symmetrically divided into IR_g+ and IR_g-. Eckard Blumschein, Magdeburg
From: Eckard Blumschein on 23 Nov 2006 03:07 On 11/23/2006 7:01 AM, Virgil wrote: > In article <45646BE7.6050307(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/22/2006 12:45 PM, mueckenh(a)rz.fh-augsburg.de wrote: >> > Virgil schrieb: >> > >> >> In article <456304B0.70705(a)et.uni-magdeburg.de>, >> >> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > Eckard>> >> > Being admittedly not very familiar with set theory, I nonetheless wonder >> >> > if sets are considered like something going on forever. >> >> Virgil>> >> Sequences do, sets do not. >> >> WM>> > Sequences are sets. >> Eckard>> Perhaps you correctly learned set theory. So I guess, William Hughes was >> correct and Virgil this time wrong. > Virgil> While all sequences are sets ( as functions from N to some set of > values) not all sets are sequences. So you could be correct if referring to sets that are no sequences? Look above: I referred to something going on forever.
From: Eckard Blumschein on 23 Nov 2006 03:11 On 11/23/2006 7:10 AM, Virgil wrote: > In article <45647BF3.7090907(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/21/2006 10:25 PM, Virgil wrote: >> > In article <4563067B.5050805(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > >> >> On 10/29/2006 10:22 PM, mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> >> > Every set of natural numbers has a superset of natural numbers which is >> >> > finite. Every! >> >> >> >> >> >> I am only aware of the unique natural numbers. If one imagines them >> >> altogether like a set, then this bag may also be the only lonly one. >> > >> > Is that "lonely"? >> >> I meant: There is only one natural sequence of numbers: 1, 2, 3, ... >> For instance 0, 1, 2, ... or 3, 4, 5, ... or 2, 4, 6, ... are something >> else. "Every" implies that there are more than one. > > There is only one sequence of all natural numbers taken in natural > order. > But there are uncountably many different sequences of natural numbers if > one is not required to include all of them nor to take them in any > particular order. Countably many, yes. As many as there are natural numbers. However, none of them is identical with the notion of _the_ natural numbers.
From: Virgil on 23 Nov 2006 03:14
In article <4565558A.2000603(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/23/2006 6:58 AM, Virgil wrote: > > In article <45646A1E.1080606(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> Virgil wrote: > >> > >> > Eckard Blumschein wrote: > >> > > >> >> William Hughes wrote: > >> >> > >> >> > As you stated "Goes on forever" is a property of the natural > >> >> > numbers. The set {1,2,3,...} is just the natural numbers. So > >> >> > this set must go on forever. > >> >> > Eckard>> >> Being admittedly not very familiar with set theory, I > nonetheless wonder > >> >> if sets are considered like something going on forever. > >> > > William>> > Sequences do, sets do not. > >> > > Virgil>> > When one represents the members of a set as members of a > sequence, as > >> > {1,2,3,...} does, that little ambiguity should not really mislead anyone. > >> > Eckard>> That little ambiguity? Doesn't it make a categorical difference > if a set > >> has been a priori set for good? > > > Virgil> When one speaks of the sequence {1,2,3,...} and another speaks > of the > > set {1,2,3,...}, the notational ambiguity should not override the words > > "sequence" and "set". > > Perhaps you did not get or at least did not accept my point: > Does it not make a categorical difference whether a set is something > unchanging beeing already perfect for good or something that just > formally includes the vital property of the series of nagtural numbers > to have no end at all? > The notation {1, 2, 3, ...} is ambiguous with this respect. It depends > on the decision whether ... denote potential or actual infinity. Are the labels "sequence" and "set" ambiguous to EB? If so perhaps he is just muddled. > > Meanwhile I see mounting evidence for my suspicion that set theory is > some sort of (self?)-deception. EB sees what he tells himself he should see, regardless of whether there is anything there to see of not. |