Cantor Confusion
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From: mueckenh on 6 Feb 2007 05:31 On 5 Feb., 04:44, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <80kcs2p92giuv70hehcbifcoild1d1u...(a)4ax.com> G. Frege <nomail(a)invalid> writes: > > > On Sun, 04 Feb 2007 21:22:31 GMT, Michael Press <rub...(a)pacbell.net> > wrote: > > > >> > > >> "Man soll den Tag nicht vor dem Abend loben." > > >> > > > I looked this up and found the meaning. > > > How does mid-sentence capitalization work? > > > > > The substantives in German are usually written this way. > > > > > Is this verse? > > > > > A saying. > > And indeed, it is also a saying in Dutch, when translated. And this was > quite appropriate as a response to my earlier posting (which it was). > But we do not have such capitals mid-sentence. In Dutch: > "Men moet de dag niet prijzen voor het avond is." Interesting. I do not speak Dutch, but I understand the full meaning: Man möge den Tag nicht preisen, bevor es Abend ist. Regards, WM
From: mueckenh on 6 Feb 2007 05:35 On 6 Feb., 11:22, Franziska Neugebauer <Franziska- Neugeba...(a)neugeb.dnsalias.net> wrote: > mueck...(a)rz.fh-augsburg.de wrote: > > Franziska Neugebauer wrote: > [...] > >> > The simplest reason is that omega - n = omega for all n in N. > > >> Where did you get that from? Reference? EB? > > > You could even figure it out by yourself. > > I cannot find any reference. Perhaps, there is none. > > F. N. > -- > xyz Already Cantor knew it, but I am too lazy to look for the reference. Read his papers, then you will encounter it. Regards, WM
From: Franziska Neugebauer on 6 Feb 2007 05:36 mueckenh(a)rz.fh-augsburg.de wrote: > On 4 Feb., 19:47, Fuckwit <nomail(a)invalid> wrote: >> On 4 Feb 2007 10:39:08 -0800, mueck...(a)rz.fh-augsburg.de wrote: [...] >> The empty set is the only set which does not have any elements > > also streng genommen als solche gar nicht vorhanden ist (Cantor) M�ckenheim axiom 3 "There is no empty set." >> Its >> existence (in ZFC) is guaranteed by the axiom of subsets (Aussonder- >> ung). You know, modern math is based on _axioms_, not some sort of >> hand waving (you seem to prefer). Usually the empty set is denoted >> with "{}". > > The existence of the empty set is not at all guaranteed. Please show. > There is an axiom which requires the existsence of the non existing Please name *that* (single) axiom. Which set theory do you refer to? > and seems to make some people happy (like the axiom which requires the > finity of the infinite). Please name that axiom which "requires the finity of the infinite". >> Then we may define (in an entirely non-circular way): >> >> 0 := {} >> 1 := {0} >> 2 := {0,1} >> 3 := {0,1,2} >> 4 := {0,1,2,3} >> >> >> >> > That is set theory, not mathematics. >> >> Well, the last time I've checked it set theory was mathematics. > > I don't believe that you are able to check what mathematics is. "What mathematics is" is not /defined/ in Augsburg. F. N. -- xyz
From: Franziska Neugebauer on 6 Feb 2007 05:42 mueckenh(a)rz.fh-augsburg.de wrote: > On 6 Feb., 11:22, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: >> mueck...(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer wrote: >> [...] >> >> > The simplest reason is that omega - n = omega for all n in N. >> >> >> Where did you get that from? Reference? EB? >> >> > You could even figure it out by yourself. >> >> I cannot find any reference. Perhaps, there is none. >> > Already Cantor knew it, but I am too lazy to look for the reference. > Read his papers, then you will encounter it. You are certainly able to brief a proof of "omega - n = omega for all n in N.", are you? And of course to define subtraction involving non-natural numbers like omega. F. N. -- xyz
From: mueckenh on 6 Feb 2007 06:27
On 5 Feb., 05:10, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > What *is* IIII. You never have defined it. You really do not like > definitions, as they pin down the real meaning. IIII is a primitive. Everybody knows it - even without definition by Peano or Dedekind. That means, we do not need Peano to know small natural numbers. > > > Henry VIII had 7 predecessors --- only including him they were 8 > > Henries. > > You are shifting position again? No. I could use IIIIIII but I can use abbreviations like VII or 7. > When I asked you about what basic > way, III c IV c V, you answered that I had to continue with IIII, > IIIII, etc. The basic way to establish IV c V is to use the numbers in their basic form IIII c IIIII. (Numbers *are* their representations.) > > > > > Correct, for instance for 1/7. > > > > > > And for computable numbers some representation does exist. > > > > But this representation does not necessarily enable us to determine > > the trichotomy relation with numbers which are really numbers. > > Perhaps. How do you establish trichotomy between 1/13 and 1/64? > Are you really going to base-26 to establish that? That would be > pathetic. But it would be possible! It would yield the famous "mathematical precision" if this could not be establised otherwise. > > > > > > > 1) 1 ist eine natürliche Zahl. > > > > > > 2) Jede Zahl a in N hat einen bestimmten Nachfolger a' in N. > ... > > > This is a recursive definition of natural numbers. By (1) we have one > > > natural number, by (2), from that single natural number we get a lot of > > > other natural numbers. > > > > Why do you say N is wrong in (2) but not in (1)? > > Where in (1) is N? I do not see N at all. "1 ist eine natürliche Zahl "means "1 in N". The property "being a natural number" implies the existence of N. Of course the requirement to decide whether n is in N is more circular than the statement that 1 is in N. > > > > This is getting phylosofical. In mathematics a definition is circular if > > > in the definition one of the deciding features is the term you want to > > > define. So a definition as you gave: > > > 3 is the set of all sets of 3 elements > > > is a perfect example of a circular definition. > > > > !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! IT IS NOT A > > DEFINITION !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! > > You stated that when I asked you for a definition. So what is happening > here? You will have read the definitions in my book. "3 is the set of all sets of 3 elements" is explained in chapter 10. > > > > > You may verify that the rings: > > > R(N, +, *) and R(K, '+', '*') > > > are isomorphic. > > > > I do know that. But I don't want some isomorphic sets. I want to > > define *the natural numbers* which are > > I > > II > > III > > ... > > And elsewhere you are using 1, 2, 3, 4, VIII, etc. Pray remain consistent. There are different forms of expressing natural numbers. I, II, III, ... is the fundamental one. 1,2,3, ... is a convenient one, in particular if numbers like 10^1000 are involved. > > > > > Whether it exists remains to be investigated. > > > > > > Your existence is not a mathematical existence. > > > > This form of existence is the only possible existence. > > As you do not define your form of existence, Read chapter 10 of my book. > it is impossible to talk about > it, at least mathematically. Present mathematics has nothing in common with existence. > > I gave two definitions: Peano and that with "+1" which is very close > > to Dedekind's attitude. (I don't know whether he actually created it, > > but I know that he would have liked it with "+1" as a primitive). But > > the axioms do not establish any existence, in particular not when you > > apply Dedekind's definition of what a number is. > > Existence is a mathematical thing when you can establish it by axioms > or through theorems based on axioms. Anything else is merely phylosophy. There is something called reality and another thing called matheology. Both are disjoint. Regards, WM |