An uncountable countable set
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From: Virgil on 30 Sep 2006 16:56 In article <1159648497.728869.269100(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1159437845.922031.117160(a)e3g2000cwe.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > > For instance: I + I = II. > > > > > > > > In some cases, yes. In other cases, no. Depends entirely on how you > > > > define "+" and the other symbols. In Greek mathematics I would expect > > > > I + I = K (as in 10 + 10 = 20). > > ... > > > > I ask for self-evident truths. Upto now you have not provided any. > > > > > > For Cantor I + I = II is such a self-evident truth. (Of course with the > > > usual meaning of "+" and "=".) > > > > How do you *define* "+"? I have not yet seen a definition from you. In > > mathematics that operation can be defined in terms of the Peano axioms. > > In *mathematics* that operation has been known and been defined well > enough 3000 years before Peano and before the cheeky set theorist. "Another" may have been defined well before that, but "plus 1" needs a coherent number system which came into being much later that "another".
From: Virgil on 30 Sep 2006 17:00 In article <1159648683.273921.24350(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1159438112.240001.268540(a)m7g2000cwm.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > > > > > The successor function *is* counting (+1). > > > > > > > > Wrong. > > > > > > After a while you will have run out of the predefined successor, > > > unavoidably. > > > > succ(x) = {x}. > > That is nothing else but a veiled form of +1. (This form of addition > of 1 is due to Zermelo's, a little bit different from that of von > Neumann's.) "Mueckenh" has it backwards, "+1" is merely a veiled form of "another", which predates counting by millennia. And both Zermelo's and von Nuemann's successor echo "another" faithfully without any requirement for "+1".
From: mueckenh on 30 Sep 2006 17:03 Virgil schrieb: > In article <1159648497.728869.269100(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > In article <1159437845.922031.117160(a)e3g2000cwe.googlegroups.com> > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > ... > > > > > > For instance: I + I = II. > > > > > > > > > > In some cases, yes. In other cases, no. Depends entirely on how you > > > > > define "+" and the other symbols. In Greek mathematics I would expect > > > > > I + I = K (as in 10 + 10 = 20). > > > ... > > > > > I ask for self-evident truths. Upto now you have not provided any. > > > > > > > > For Cantor I + I = II is such a self-evident truth. (Of course with the > > > > usual meaning of "+" and "=".) > > > > > > How do you *define* "+"? I have not yet seen a definition from you. In > > > mathematics that operation can be defined in terms of the Peano axioms. > > > > In *mathematics* that operation has been known and been defined well > > enough 3000 years before Peano and before the cheeky set theorist. > > "Another" may have been defined well before that, but "plus 1" needs a > coherent number system which came into being much later than "another". One needs no system at all. Make some IIII and in order to add one, make one more IIIII. You need not even to know a name for one of these sets. Compare Lorenzen. Regards, WM
From: Virgil on 30 Sep 2006 17:08 In article <1159649021.675137.307470(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > Indeed. But when it is *defined* as the limit of a sequence, and if that > > limit exists, that means that the object does exist. > > The limit omega does not exist. Then the number one does not exist either. > A set can be a function of time. Only for a constant function. If the memberships are different, the sets are different, a la the axiom of extentionality. So the codomain of such a "function" must be a set of sets, one for each time in the domain of the function.
From: Dik T. Winter on 30 Sep 2006 22:26
In article <1159648497.728869.269100(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1159437845.922031.117160(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > > For instance: I + I = II. > > > > > > > > In some cases, yes. In other cases, no. Depends entirely on how you > > > > define "+" and the other symbols. In Greek mathematics I would expect > > > > I + I = K (as in 10 + 10 = 20). > > ... > > > > I ask for self-evident truths. Upto now you have not provided any. > > > > > > For Cantor I + I = II is such a self-evident truth. (Of course with the > > > usual meaning of "+" and "=".) > > > > How do you *define* "+"? I have not yet seen a definition from you. In > > mathematics that operation can be defined in terms of the Peano axioms. > > In *mathematics* that operation has been known and been defined well > enough 3000 years before Peano and before the cheeky set theorist. Whatever you say. Pray stay in your 3000 years old mathematics. Nathenatics has been concerned with axiomatic definitions on what they were doing since, I think, Euclides. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |