Cantor Confusion
in [Math]
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From: Dik T. Winter on 15 Feb 2007 22:39 In article <gn2at2565gmfg5kjlbpr2jn10l6mfuiht3(a)4ax.com> G. Frege <nomail(a)invalid> writes: > On Thu, 15 Feb 2007 20:29:22 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: .... > >>> For one thing, WM is missing the knowledge that a name is not the object > >>> named. > >>> > >> Is this valid for names too? > >> > > I think so. My name is "David". ""David"" is a name for my name. Another > > name for my name would be "my name". > > > Actually, again one of the many insights of Gottlob Frege [the real > one]. Actually it has already been used in Alice in Wonderland, published when Frege was 17 years old: Knight: The name of the song is called Haddocks' Eyes. Alice : Oh, that's the name of the song, is it? Knight: No, you don't understand, that is wat the name is called, the name really is The Aged Aged Man. Alice : Then I ought to have said that's what the song is called? Knight: No, you oughtn't: that's quite another thing! The song is called Ways And Means, but that's only what it's called. Alice : Well, what is the song then? Knight: I was coming to that, the song really is A-sitting On A Gate. Lewis Carroll was not entirely stupid... -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 15 Feb 2007 23:09 Dik T. Winter wrote: > In article <gn2at2565gmfg5kjlbpr2jn10l6mfuiht3(a)4ax.com> G. Frege <nomail(a)invalid> writes: > > On Thu, 15 Feb 2007 20:29:22 -0500, David Marcus > > <DavidMarcus(a)alumdotmit.edu> wrote: > ... > > >>> For one thing, WM is missing the knowledge that a name is not the object > > >>> named. > > >>> > > >> Is this valid for names too? > > >> > > > I think so. My name is "David". ""David"" is a name for my name. Another > > > name for my name would be "my name". > > > > Actually, again one of the many insights of Gottlob Frege [the real > > one]. > > Actually it has already been used in Alice in Wonderland, published when > Frege was 17 years old: > Knight: The name of the song is called Haddocks' Eyes. > Alice : Oh, that's the name of the song, is it? > Knight: No, you don't understand, that is what the name is called, the > name really is The Aged Aged Man. > Alice : Then I ought to have said that's what the song is called? > Knight: No, you oughtn't: that's quite another thing! The song is > called Ways And Means, but that's only what it's called. > Alice : Well, what is the song then? > Knight: I was coming to that, the song really is A-sitting On A Gate. > Lewis Carroll was not entirely stupid... As Martin Gardner pointed out, the Knight should have just started to sing the song after saying "... the song really is". "A-sitting On A Gate" is another name, not the song itself. -- David Marcus
From: G. Frege on 15 Feb 2007 23:07 On Fri, 16 Feb 2007 03:39:36 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: > > Actually it has already been used in Alice in Wonderland, published when > Frege was 17 years old ... > Actually I was referring to (1) the insight that there really a problem is lurking and (2) the idea of using quotation marks to prevent such confusion. :-) See: http://plato.stanford.edu/entries/quotation/ F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 15 Feb 2007 23:19 On Fri, 16 Feb 2007 05:07:47 +0100, G. Frege <nomail(a)invalid> wrote: >> >> Actually I was referring to (1) the insight that there really a >> problem is lurking and (2) the idea of using quotation marks to >> prevent such confusion. :-) >> > See: > http://plato.stanford.edu/entries/quotation/ > If you read texts by CANTOR and/or DEDEKIND you will see that they simply fail to make any _visible_ difference between "using" a symbol and "mentioning" it. :-( Just a simple example, CANTOR writes: _ V3 ist also nur ein Zeichen f�r eine Zahl, welche erst noch gefunden werden soll, nicht aber deren Definition." _ [So V3 is is just a symbol for a number, which <etc.>] But this is just plain nonsense (if taken "literally"). FREGE would have written: _ "V3" ist also nur ein Zeichen f�r eine Zahl, <etc.> _ [So "V3" is is just a symbol for a number, which <etc.>] Actually, this habit (NOT to use quotation marks) is still quite common among mathematicians (i.e. in mathematics in general). :-( F. -- E-mail: info<at>simple-line<dot>de
From: mueckenh on 16 Feb 2007 03:38
On 15 Feb., 14:05, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1171469941.993167.166...(a)h3g2000cwc.googlegroups.com> mueck....(a)rz.fh-augsburg.de writes: > > > On 14 Feb., 02:22, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > > > > Numbers can express properties? You have lost me here. > > > > > > > > > > > > To have three elements is a property of a set. > > > > > > > > > > Oh. Your terminology is unfathomable. Indeed the number 3, when seen > > > > > as a set, can have three elements. In the von Neumann model. However, > > > > > I remember to also having seen another model, where the number three was > > > > > {{{}}} > > > > > > > > It was page 93 of my book. > > > > > > I have seen it earlier than that. > > > > By the way, above is only number 2 given. > > Indeed, my error. So your comment: > > When seen as a set of curly brackets it has 3 at the left sinde and 3 > > at the right. > was actually completely irrelevant. Let's get on with the actual > representation of 3: {{{{}}}}. This is not a representation of 3 other than in a perverted system, which calls 0 the first number, 1 the second and so on. Of course {{{{}}}}, or better and easier {{{{, denotes the fourth number which is 4 and not 3. Only set theory needs this absurd definition of nought to be the "first" number, because 3 counting the numbers up to 2 but not 3 itself fits well with omega counting all the natural numbers but being not a natural number. No, this is sham does not help: The number of all natural numbers´, if existing, is a natural number, because the natural numbers count themselves (so they were designed). |{1}| = 1 |{1,2}| = 2 |{1,2,3}| = 3 .... |{1,2,3,...}| = ... i.e. potentially infinite, not fixed, capable of growing without bound, denoted by oo but not by a fixed number omega. Regards, WM |