Virgil
Posts:
7,021
Registered:
1/6/11


Re: Mathematics in brief
Posted:
Dec 8, 2012 6:58 PM


In article <4754d1053a554373a304cdc84aaa6991@4g2000yqv.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> On 8 Dez., 18:55, Zuhair <zaljo...@gmail.com> wrote: > > On Dec 8, 1:49 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > > > > > > > > > > On 8 Dez., 10:41, Zuhair <zaljo...@gmail.com> wrote: > > > > > I will give two answers, the first one depends on the finity of the > > > universe, the second one does not. > > > > > > Mathematics is "discourse > > > > > and discourse needs the tools available to us. They all are finite. > > > > > > How those forms are known to us? the answer is through their > > > > exemplification as part of the discourse of consistent theories about > > > > form. > > > > > Consistent means that discourse about all elements is possible. > > > We know that all finite words belong to a countable set. We know that > > > no infinite word can be mentioned without having a finite name. > > > > > Therefore this silly argument is really silly: > > > "I can decide for a real number x whether a real number y deviates in > > > its decimal (or any other) expansion from that of x." > > > The complete infinite expansion of x is never known, not even in an > > > infinite universe, but only the finite formula allowing expansion to > > > any required level. > > > > > So the argument is silly that there are uncountably many x because x > > > has an infinite expansion. No x is known without a finite formula, > > > name, word, ... > > > > There is no known inconsistency with > > uncountability > > I know an inconsistency.
You claim to know one, but, as usual, your claims far exceed your abilities to confirm those claims.
> And many others know it too. Only people who > refuse to know it will never know it.
Which is our valid comment about what WM rejects. > > For all nonmatheologians this is clear: All reals of the unit > interval are represented as paths in the Binary Tree. All > distibguishable paths of the Binary Tree belong to a countable set > because the complete tree can be covered by countably many paths.
Provably false, since every attempt at covering set of paths in the tree can be shown, via a simple diagonal proof, to be incomplete.
Thus any claim to completeness necessarily fails.
> So > there is no chance to distinguish further paths (like the famous 1/3) > by nodes.
The issue is not whether you can distinguish more of them but whether you can list all of them. Since it is trivial to prove that the set of reals is at least countable (N can be injected into R) it only remains to consider whether some such injection can be a surjection.
Which it provably cannot!!!!
> It can only be distinguished by a finite deifnition like > "1/3" or "0.010101...". This makes uncountability inconsistent  for > thinkers who do not refuse to take notice.
It has nothing at all to do with countability, which is entirely an issue of whether some injection from N to R, or other mapping from N to R, can be surjective.
Cantor provided two quite different proofs that no such surjections can exist, neither of which has been successfully refuted, so that, as things stand now, R is still uncountable. > > > Mathematics only > > needs to speak about possible forms. > > Uncountable sets are impossible forms.
Sets are defined by their criteria for membership, not necessarily by listability of their members, though listing is a common way of defining small sets.
WM seems to think that any set that is not (or, at least, cannot be) defined by a listing of its members does not exist.
But sets are defined by their criteria for membership, not by the listability of their memberships. > > > Now whether it is TRUE that we > > have uncountably many reals in the real world, this is another matter > > that belong to applied mathematics and not to pure mathematics. > > Uncountability is already impossible in the ideal world. It is impossible in a physical world, but not in a mental one.
I repeat: In mathematics a set is defined by an unambiguous test for membership in it, and that need not, and often does not, involve any listing any of its members at all.
> You can > believe that you can think of uncountable sets. But you cannot > distinguish so many elements in principle.
But sets are not made that way, sets are created by their membership criteria, not necessarily by listing, or distinguishing, their members.
So what WM is talking about is the theory of listable sets with distinguishable members, not the mathematical theory of sets, which includes a lot of sets outside his theory.
>So your belief is unfounded > and nonmathematical.
On the contrary, it is your version of set theory which is not mathematical as it excludes many sets which mathematical set theories include. Any rule which unambiguously separates those things which satisfy some criterion from those that do not, defines and creates a set of those things which do satisfy that criterion.
That such a definition creates sets which one can prove to be uncountable is obvious to mathematicians, however much WM bitches about it.
> That has nothing to do with reality but with the > countability of all finite words. In the real world you could not even > distinguish more than 10^100 elements.
A mathematical set only has to distinguish between elements and nonelements, never between one of its element and another.
That WM's sets seem to have additional requirements imposed upon them is mathematically irrelevant to any truly mathematical set theory. > > Regards, WM 

