Cantor Confusion
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From: William Hughes on 14 Oct 2006 04:41 mueckenh(a)rz.fh-augsburg.de wrote: > imaginatorium(a)despammed.com schrieb: > > > David Marcus wrote: > > > Dik T. Winter wrote: > > > > In article <1160649760.068206.172400(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > > To inform the set theorist about the possible existence of sets with > > > > > finite cardinality but without a largest number. > > > > > > > > Interesting but in contradiction with the definition of the concept of > > > > "finite set". So you are talking about something else than "finite > > > > sets". > > > > > > It would seem he is. I don't understand why people use words in non- > > > standard ways without explaining what they mean. They are guaranteeing > > > that no one will understand them. > > > > Several possible obvious answers. (BTW, it was fairly clear you were > > new around here -- then you tried asking Ross Finlayson what he means. > > Clinched it.) > > > > (a) The writer is playing a bizarre game of trollery. > > (b) The writer is simply misinformed about the meaning of a particular > > term. (Don't think this is common) > > (c) The writer does not have the mental apparatus to understand a > > formal argument, and therefore simply cannot comprehend the difference > > between a number of statements of subtle difference. This seems to be > > most common. For example, Mueckenheim - who astonishingly appears to > > *teach* mathematics at some sort of college in Germany > > University of Applied Sciences, Augsburg. > > - plainly cannot > > comprehend the difference that swapping quantifiers makes. He cannot > > comprehend that there might be a difference between the significance of > > "every" in "Every girl in the village has a lover" and "John makes love > > to every girl in the village". > > Is the Imaginator too simple minded to understand, or is it just an > insult? The quantifier interchange is impossible in general, but it is > possile for special *linear* sets in case of *finite* elements. > It is true that in some cases the quantifier exchange is possible. However, the fact that the quantifier exchange is impossible in general means that you cannot use quantifier exchange in a proof without explicit justification of the step. Despite you protestations, this is exactly what you do. - William Hughes
From: William Hughes on 14 Oct 2006 04:44 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > Dik T. Winter wrote: > > > In article <1160649760.068206.172400(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > To inform the set theorist about the possible existence of sets with > > > > finite cardinality but without a largest number. > > > > > > Interesting but in contradiction with the definition of the concept of > > > "finite set". So you are talking about something else than "finite > > > sets". > > > > It would seem he is. I don't understand why people use words in non- > > standard ways without explaining what they mean. They are guaranteeing > > that no one will understand them. > > A finite set is a set with a number of elements, which is smaller than > some natural number. As far as I know this notion is covered by the > standard meaning of words. > > I talked about a set with less than 100 elements. Therefore I used the > word "finite set". > No, you talked about a something that was not a set. You used the term "finite set" in a non-standard way, but the part that was non-standard was "set" not "finite". - William Hughes > Regards, WM
From: William Hughes on 14 Oct 2006 04:48 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160675140.906009.253460(a)i42g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > this implies > > > > > > > > for every digit position N, > > > > there exists a single unary number, M, > > > > such that M covers 0.111... to position N > > > > > > > > > > > > this does not imply > > > > > > > > there exists a single unary number M such that for every digit > > > > position N, M covers 0.111... to position N > > > > > > Why shouldn't it? > > > > In general > > "for all x there is a y such that f(x,y)" > > does not imply > > "there is a y such that for all x f(x,y)". > > > > To establish the latter requires proof over and above the former. > > I did not state that this be true in general, but it is true in a > special case, namely for the covering of linear sets of finite > elements. > Your putative proof of this "fact" depends on a step in which quantifiers are switched without justification. - William Hughes > Regards, WM
From: William Hughes on 14 Oct 2006 05:01 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > Han de Bruijn wrote: > > > Dik T. Winter wrote: > > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > ... > > > > > It is not > > > > > contradictory to say that in a finite set of numbers there need not be > > > > > a largest. > > > > > > > > It contradicts the definition of "finite set". But I know that you are > > > > not interested in definitions. > > > > > > Set Theory is simply not very useful. The main problem being that finite > > > sets in your axiom system are STATIC. They can not grow. Which is quite > > > contrary to common sense. (I wouldn't imagine the situation that a table > > > in a database would have to be redefined, every time when a new row has > > > to be inserted, updated or deleted ...) > > > > Is your claim only that set theory is not useful or is contrary to > > common sense? Or, are you claiming something more, e.g., that set theory > > is mathematically inconsistent? > > It is not useful and contrary to common sense These are opinons. > but above all it is > mathematically inconsistent. This requires proof. Try to produce one without any unjustified quantifier exchange. - William Hughes
From: William Hughes on 14 Oct 2006 09:36
mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160648741.707624.62340(a)m7g2000cwm.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > In article <1160577085.758246.228800(a)e3g2000cwe.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > If discontinuous functions were easily allowed everywhere, why then do > > > > > you think that > > > > > lim{n-->oo} n < 10 > > > > > or > > > > > lim{n-->oo} 1/n > 10 > > > > > would be wrong? > > > > > > > > Since N is not normally considered to be a topological space, continuity > > > > of functions with a non-topological domain N is a contradiction in > > > > terms. > > > > > > Apply your knowledge to the balls of the vase. > > > > Which knowledge tells me that at noon each and every ball has been > > removed from the vase. > > You are joking? > > > > > > > > On the other hand, limits of real sequences (functions from N to R) have > > > > been quite adequately defined. One such definition is: > > > > Give f:N --> R and L, then > > > > lim_{n in N} f(n) = L (or lim_{n --> oo} f(n) = L > > > > is defined to mean > > > > For every real eps > 0, Card({n: Abs(f(n)-L) > eps}) is finite. > > > > > > For the vase problem with the number n(t) of balls in the vase after t > > > transactions we can find always a positive eps such that for t > t_0: > > > 1/n(t) < eps, hence n(t) larger than an arbitrary positive number. > > > > But that analysis does not carry beyond the times of transition, and > > those times do not include noon or go past noon. > > > > You are assuming properties not given. > > If we can meaningfully calculate the limit of the harmonic sequence > 1, 1/2, 1/3, ... --> 0 > or the sum of the geometric series > 1 + 1/2 + 1/4 + ... --> 2, > then we can also calculate that at noon 1/n = 0. > > > > > > > Therefore, your assumption of lim {t-->oo} n(t) = 0 is absurd. > > > > Your assumption that some ball that has been removed has not been > > removed is even more absurd. > > Yes, that is true. Therefore the existence of all natural numbers can > be excluded. To assume it is absurd. > No, the conclusion is that assuming the existence of all natural numbers leads to counterintuitive results. Absurditiy is just an opinion. If you do not want to assume the existence of all natural numbers, knock youself out. But the fact that you don't like the counterintuitive results does not mean that assuming the existence of all natural numbers is contradictory. - William Hughes > Regards, WM |