Cantor Confusion
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From: Ralf Bader on 21 Jan 2007 00:54 G. Frege wrote: > On Sat, 20 Jan 2007 18:21:06 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > >> >> mueckenh(a)rz.fh-augsburg.de wrote: >>> >>> The tree illustrates the following problem: >>> >>> 1) For every natural number n there is an natural number m >>> such that m = n+1. >>> >>> In short, with n,m in N: An Em: m = n+1 >>> >> So, what's the problem [...]? >> > The problem is that WM suffers from quantifier dyslexia. > > From > > An e N Em e N: m = n+1 > > he concludes (correctly) > > An e N Em e N: m > n. > > And in his understanding this means (is logically equivalent to): > > * Em e N An e N: m > n. Mückenheim is an avid reader of Cantor's original papers, but he doesn't understand them (I could give examples, but one must know German for this). It probably wasn't one of Cantor's better ideas to use notations like (n) for the set of natural numbers, i.e. a set is denoted by an adorned variable taking values from that set. It may come from there that, for Mückenheim, a variable like n seems to denote any natural number at any time. If one allows variables to vary that much, then this leads to nonsense like the above in a quite natural way. > He's not able to grasp that AxEy(phi[x,y]) isn't equivalent to EyAx > phi[x,y] in general. > > A quote from Mückenheim's oeuvre: > > <begin quote> > > ...it is inconsistent to speak of /an infinite set of finite numbers/. > Finite numbers can only form a potentially infinite set. An actually > infinite set cannot exist other than including its cardinal number > aleph_0 or the number "ordinal infinity", denoted by 'omega' This had > been unconsciously acknowledged by Cantor himself already: "/Every > number/ smaller than omega is a finite number, and its magnitude /is > surpassed by other finite numbers/." Here the phrase "by other finite > numbers" is obviously to be interpreted as "by such finite numbers > which did not yet belong to the set containing /every finite number/". > > <end quote> If the set of natural numbers is denoted (n), and m is a natural number greater than n then... Ralf
From: David Marcus on 21 Jan 2007 01:32 G. Frege wrote: > A quote from M�ckenheim's oeuvre: > > <begin quote> > > ...it is inconsistent to speak of /an infinite set of finite numbers/. > Finite numbers can only form a potentially infinite set. An actually > infinite set cannot exist other than including its cardinal number > aleph_0 or the number "ordinal infinity", denoted by 'omega' This had > been unconsciously acknowledged by Cantor himself already: "/Every > number/ smaller than omega is a finite number, and its magnitude /is > surpassed by other finite numbers/." Here the phrase "by other finite > numbers" is obviously to be interpreted as "by such finite numbers > which did not yet belong to the set containing /every finite number/". > > <end quote> The word "obviously" is a nice touch. Always good to toss in a few "obviously"'s when saying things that are wrong or nonsensical. -- David Marcus
From: G. Frege on 21 Jan 2007 01:43 On Sun, 21 Jan 2007 06:54:30 +0100, Ralf Bader <bader(a)nefkom.net> wrote: > > It probably wasn't one of Cantor's better ideas to use notations like > {n} for the set of natural numbers, i.e. a set is denoted by an adorned > variable taking values from that set. > Agree, though one might read it as an abbreviation of {n : n is a natural number} , and n is just a "typical" member of that set (sort of). > > It may come from there that, for M�ckenheim, a variable like n seems > to denote any natural number at any time. > Imho, the main reason is his general lack of mathematical understanding: (imho) he's just a mathematical "illiterate". Of course this makes it even more difficult (i.e. quite impossible) to understand Cantor correctly. And of course that's the reason why he's raving that vigorously against "Cantor" (in particular) and set theory (in general). [Not taking the less obvious psychological reasons into account here.] F. -- E-mail: info<at>simple-line<dot>de
From: David Marcus on 21 Jan 2007 02:08 Ralf Bader wrote: > Mueckenheim is an avid reader of Cantor's original papers, but he doesn't > understand them (I could give examples, but one must know German for this). > It probably wasn't one of Cantor's better ideas to use notations like (n) > for the set of natural numbers, i.e. a set is denoted by an adorned > variable taking values from that set. It may come from there that, for > Mueckenheim, a variable like n seems to denote any natural number at any > time. While a person might be confused by such notation, anyone with any sense would quickly get back on track once their error was pointed out. Mueckenheim's problems clearly go very deep. The less people know, the less they know what they know (and that's the sane ones). -- David Marcus
From: G. Frege on 21 Jan 2007 02:07
On Sun, 21 Jan 2007 01:32:15 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> >> A quote from M�ckenheim's oeuvre: >> >> <begin quote> >> >> ...it is inconsistent to speak of /an infinite set of finite numbers/. >> Finite numbers can only form a potentially infinite set. An actually >> infinite set cannot exist other than including its cardinal number >> aleph_0 or the number "ordinal infinity", denoted by 'omega' This had >> been unconsciously acknowledged by Cantor himself already: "/Every >> number/ smaller than omega is a finite number, and its magnitude /is >> surpassed by other finite numbers/." Here the phrase "by other finite >> numbers" is obviously to be interpreted as "by such finite numbers >> which did not yet belong to the set containing /every finite number/". >> >> <end quote> > > The word "obviously" is a nice touch. Always good to toss in a few > "obviously"'s when saying things that are wrong or nonsensical. > Indeed! (Since it allows for very smooth "argumentation" concerning crank stuff. I mean, it is really helpful when dealing with things which might interrupt your flow of ...well... "thoughts".) Another one in that vain: "As there is only a countable set of intervals of length 1 in R, the denumerability of the whole set R is obvious." (WM, The Meaning of Infinity) Obvious, indeed. F. -- E-mail: info<at>simple-line<dot>de |