Cantor Confusion
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From: William Hughes on 8 Dec 2006 18:45 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > Recall this post from Dec 1 > > > > We extend this to potentially infinite sets: > > > > A function from the potentially infinite set A to the > > potentially infinite set B is a potentially infinite set of > > ordered pairs (a,b) such that a is an element of A and b is > > an element of B. > > A function, according to modern mathematics, is a set, actually fixed > and complete. The expression "variable" is merely a relict from ancient > times when people knew that the objects of mathematics do not exist in > some nirvana but have to be present in a mind where not everything can > be present simultaneously. > > > > > We can now define bijections on potentially infinite sets > > Only if we consider them being actually infinite. But that would > exclude them from being potentially infinite. > Given that you find the words function, and bijection so distasteful, I have reworked my points to avoid using them in conection with potentially infinite sets. Please stop me at the first point you disagree with. -a potentially infinite set exists (this leaves open the question of whether the elements of a potentially infinite set exist.) -if we are given x and a potentially infinite set we can determine whether x is an element of the potentially infinite set. -if we are given two potentially infinite sets A and B, we can contstuct a third potentially infinite set C, consisting of ordered pairs (a,b) where a is an element of A, and B is an element of B. - definitions a transform between two potentially infinite sets A and B, or between a potentially infinite set A and a set B is a potentially infinite set of ordered pairs (a,b) where a is an element of A, and B is an element of B. a transform between two sets G and H is a function. an equitransform between two potentially infinite sets A and B or between a potentially infinite set A and an set B is a potentially infinite set, C, of ordered pairs (a,b) where a is an element of A, and B is an element of B. and if a_1 and a_2 are different (a_1,b_1) and (a_2,b_2) are elements of C then b_1 and b_2 are different if b_3 is an element of B then there exists a_3 and elment of A such that (a_3,b_3) is an element of C (note an equitransform is a transform) an equitransform between two sets G and H is a bijection. -given two potentially infinite sets A and B the question "Is there an equitransform between A and B?" has an answer which exists. -the equitransform defines an equivalence relation, call this relation equitransformity -a swallow is an equivalence class on sets and potentially infinite sets with respect to the equivalence relation equitransformity. - For a set A the swallow of A is the cardinal number of A. -A and B have the same swallow iff there is an equitranform between A and B. -given a potentially infinite set A, the potentially infinite set C of ordered pairs (a,a) exists, where C has the property if a is an element of A then (a,a) is an element of C Call C the identity transform. C is an equitransform on A. -A belongs to an equivalence class with respect to the equivalence relation equitransformity -A has a swallow -the swallow of A is not a natural number -if F is a set of natural numbers with a largest element n, then the swall of of F is a natural number, less than or equal to n -given two sets of natural numbers E and F where E is a potentially infinite set, and F has a largest element. there does not exist an equitransform between E and F -the diagonal is the potentially infinite set of natural numbers. -every line L has a largest element -there is no equitransform between the diagonal and a line L - William Hughes
From: Virgil on 8 Dec 2006 19:05 In article <1165614850.907256.254500(a)j72g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > [concerning Cantor's first proof of uncountability of the real numbers] > > > > In the reals, any subset which has a real upper bound has a real least > > upper bound and, similarly , any subset which has a real lower bound has > > a real greatest lower bound. > > > > The only subsets of the reals for which there is a similar property are > > real intervals. > > > > Thus it is only for the set of all reals, or for real intervals, that > > the proof appplies. > > Yes. And it does not apply if only one single element of the > investigated real interval is missing. As the uncontability property of > this interval cannot depend on this single element, the whole proof > fails. The proof does not fail. It merely does not apply to the whole any more, but it does apply separately to each of the pieces separated by that removed point. Or is WM arguing that removing a single element from a set can make an uncountable set countable, even though its removal real partitions the remaining reals into two equally uncountable sets. > > > > On the other hand, the proof can show the uncountability of a countable > > > set. If, for instance, the alternating harmonic sequence > > > (-1)^n/ n --> 0 > > > is taken as sequence (1), yielding the intervals (-1 , 1/2), (-1/3 , > > > 1/4), ... we find that > > > its limit 0 does not belong to the sequence, although the set of > > > numbers involved is obviously > > > denumerable. > > > > That doesn't work it unless you show that NO sequence can be made > > including all of the values of S = {0} union { (-1)^n/n: n in N} > > And Cantor's proof doesn't work unless one can show that there is a > missing number in every sequence. But just this cannot be done for a > real interval with only one real number missing. > > Regards, WM
From: Virgil on 8 Dec 2006 19:07 In article <1165615065.285043.115990(a)73g2000cwn.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > 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. > > > A function is a set of ordered pairs and as such it is not variable. So sets of ordered pairs cannot be 'variable' but other sets can? > The expression "variable" is merely a relict from ancient times when > people knew that the objects of mathematics do not exist in some > nirvana but have to be present in a mind where not everything can be > present simultaneously. > > Regards, WM
From: Virgil on 8 Dec 2006 19:11 > William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > A function, according to modern mathematics, is a set, actually fixed > and complete. The expression "variable" is merely a relict from ancient > times when people knew that the objects of mathematics do not exist in > some nirvana but have to be present in a mind where not everything can > be present simultaneously. > > > > > We can now define bijections on potentially infinite sets > > Only if we consider them being actually infinite. But that would > exclude them from being potentially infinite. That blows all arithmetic, at least in WM-land as he cannot define functions like addition or multiplication on merely potentially infinite sets of numbers.
From: Dik T. Winter on 8 Dec 2006 19:01
In article <5pejn2d7ekq7stdqbp8cg31ukse5mlnka6(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: .... > Of course. I was just mocking David's idiotic definition of definition > as only an abbreviation which he doesn't seem particularly anxious to > defend. I suppose you did not understand him (as you do not understand many people). A definition provides a short term for a long sentence, or even more than a single sentence. That is when I write: Let N be the set of natural numbers that is a definition of N as a short-hand for "the set of natural numbers". Note also *what* is defined here: N. So when somebody asks for a definition of "protential infinite" the question is for a long description of what that is. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |