Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Franziska Neugebauer on 6 Dec 2006 11:21 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Dik T. Winter schrieb: >> >> > Everybody knows what the number of ther EC states is. >> [...] >> > The number of EC states is "the number of EC states". >> >> This is hardly a definition. >> >> > It is simply a notion which can be equal to a natural number. >> >> Which may _evaluate_ to a number. > > No. It evaluates to a number as little as 6 evaluates to a number. It > *is* a number, though not a fixed number. Mathematically one modells such "not-fixed numbers" as functions. Conclusively this function has value 6 at 1968. > That is a matter of definition of the word "number". Provide one. Don't forget to provide a definition of "not-fixed" number and "not-fixed" set. And please show that one gains advantage over the function concept. >> Without explicitly or implicitly >> providing a context (year) there is no definite answer in terms of >> natural numbers. Mathematically the number of EC states can be >> modelled as _function_ of time. >> >> > The set of prime numbers does not contain the number 1. >> >> According to a widespread defintion of "prime number" the set of >> prime numers _refers_ _to_ a set which does not contain the number 1. >> >> > But once upon a time it did contain it. >> >> There may have been a time, when _a_ _different_ _set_ (one >> containing the 1) was _referred_ _to_ _by_ the named "set of prime >> numbers". > > That is a matter of definition. It was *the* set of prime numbers. But > it is really superfluous to reply that, according to your definition, > it was another set. Of course it was another set, because the set of > prime numbers has changed. You should take Virgil's hint to Korzybski seriously. >> This does not imply that the set of former times "has changed" in >> time. Only the naming has changed due to a changed definition. You >> may compare this with Gerhard Schr�der who did not undergo a gender >> transformation when Angela Merkel became chancellor in 2005. >> > But "the chancellor" did. I would think most people start laughing at the questioner when asked whether In 2005 the chancellor underwent a gender transformation. is true. F. N. -- xyz
From: Bob Kolker on 6 Dec 2006 11:25 mueckenh(a)rz.fh-augsburg.de wrote: > > One of many examples: The set {2,4,6,...,2n} has a cardinal number less > than some numbers in the set. This does not change when n grows (yes, > it can grow!) over all upper bounds. Therefore the assertion that the > set of all even natural numbers has a cardinal number gretaer than any > even number is false. The set of natural numbers has a cardinal greater then any set {1, 2, ... , n} for any integer n. Also the set of natural numbers has a cardinality greater than the cardinality of any set {2*1, 2*2 ..., 2*n} for any integer n. Since the set of integers is equinumerous with the set of even integers (by way of the mapping n<->2*n) it follows that the cardinality of the set of even integers is greater than the cardinality of the (finite) set {2*1, 2*2, ... ,2*n} for any integer n. You have reasoned incorrectly. So what else is new? Bob Kolker
From: Franziska Neugebauer on 6 Dec 2006 11:39 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Bob Kolker schrieb: >> [...] >> >> Rational numbers are non-genuine. Nowhere in the physical world >> >> outside of our nervouse systems do they exist. >> > >> > The nervous system is a part of the physical world. >> >> Hence this issue is to be disussed in a physics or neuro sciences >> newsgroup. >> >> > The integers belong to the nervous system. >> >> The nervous system is generally not an issue in mathematics. > > That is why many mathematicians overestimate their capabilities so > grossly. Which mathematician overestimates her or his capabilities? >> > They would not exist without the nervous system. >> >> This is a miscomprehension. Mathematical numbers exist >> mathematically. > > And nobody knows what has to be understood by this sentence. How do you know? You never have asked anybody. >> If you want to discuss how abstract entities, i. e. naive numbers, >> "exist" in the sense of "are represented in the neural system" >> you should discuss this issue in an appropriate neuro sciences >> newsgroup. > > New facts about the foundations of mathematics should be appropriate > in a math news group. Objection. Representation of abstract entities in the neural system is in general inappropriate in a math news group. You may discuss a theory of that representation in part in a math news group. But you have yet presented hardly anything which would deserve the name "theory". >> > But some people nevertheless believe in the immortal soul. >> >> This is not a mathematical issue. > > It is a parallel. The belief in the inexistent: Soul - infinty. There is a parallel between your and Hermann G�ring's vacation plans. >> > I think that is just as strange as the belief in one's >> > own capability of imaging actual infinity. >> > And, remarkably, the creator of this theory believed in both, the >> > soul/God and actual infinity, >> >> It is remarkable that the former Generalfeldmarschall of Nazi Germany >> Hermann G�ring frequently spent his vacation in the Harz. Just like >> you (and possibly Cantor, too,) do. To me it is not clear whether you >> prefer to wander the trails of Cantor or that of G�ring. > > What has that to do with mathematics? Got it now? >> > trying to prove the latter by the existence of the former. >> >> Since mathematically there is not a "god" this "proof" was hardly a >> mathematical one. > > It was. Believing in axioms and believing in God is closely connected. Prefering the same location for vacation exposes a closely connected mentality. F. N. -- xyz
From: Franziska Neugebauer on 6 Dec 2006 11:41 mueckenh(a)rz.fh-augsburg.de wrote: > A "fact" of pure tought can be reduced to and understood by > neuroscience. Reduce it in sci.neurosciences F. N. -- xyz
From: Eckard Blumschein on 6 Dec 2006 12:21
On 12/6/2006 12:55 AM, Ralf Bader wrote: > Bob Kolker wrote: > >> Eckard Blumschein wrote: >>> an exact numerical representation available. Kronecker said, they are no >>> numbers at all. Since the properties of the reals have to be the same as >>> these of the irrationals, all reals must necessarily also be uncountable >>> fictions. >> >> For the latest time. Uncountability is a property of sets, not >> individual numbers. There is no such thing as an uncountable real >> number. Nor is there any such thing as a countable integer. Countability >> /Uncountability are properties of -sets-, not individuals. >> >> You have been told this on several occassions and you apparently are too >> stupid to learn it. > > These people (Blumschein and Mückenheim) don't know how words and notions > are used in mathematics. Blumschein takes a word, e.g. "uncountable", and > attempts to infer from its everyday usage its mathematical meaning. That > such-and-such is uncountable means for Blumschein that it doesn't have the > nature of a number, or something like that. So integers are "countable" for > Blumschein, whereas irrationals are swimming in that sauce Weyl spoke about > (but not meaning what Blumschein thinks) and therefore are "uncountable" in > Blumscheins weird view. For Blumschein, your explanations are just your > prejudices. It is pointless to repeat them. He can't understand you. According to the axiom of extensionality, a set has been determined by its elements. Why do you not admit the possibility that countability of a set requires countable numbers. Doesn't it make sense? I didn't find a single counter-example. > > > Ralf |