Matt Grime said:
Your definition of 'all' is garbage.
I told it to ahrkron and also I tell you, my work is like a mirror at the first stage.
If you understand it you can see beyond your own reflection, but in your case you see nothing but your face in it.
Matt Grime said:
If S is any (set) theory in which there is a model of the natural numbers (you've never managed to include that), then there exist a proposition, P, such that both P and not P are consistent with S.
Since you cannot go beyond the standard formal definitions, you cannot understand that I am talking about
x_AND_not_
x which are members of set S of 'any undefined element of some (set) theory'
where S is well defined in this (set) theory.
What we get in this case is a well-defined set S as a 'Trojan horse' which includes in it elements that cannot be defined in the framework of the examined theory.
So as you see i do not stay in the original formal definition of Godel's theorem, but I use the deep principle of it to develop another point of view, which is much more interesting (in my opinion) then Godel's theorem.
In short, Godel's theorem is often used to show the limitations of a system, but my point of view is to show that any limited system actually lead us to search beyond its domain, which is a positive approach of the same idea.
Since you do not aware to the power of the philosophical thinking as a profound tool for deep mathematical fundamental ideas, you cannot see but the reflection of your formal face in my philosophical mirror.
Matt, if you think that a good mathematician is a walking encyclopedia of formal mathematical knowledge, which is used to produce results within a particular brunch of the standard framework, and he never think beyond his own domain, then your way is not my way.
------------------------------------------------------------------------------------------------
Matt Grime said:
Your definition of 'all' is garbage.
Theorem: Matt's response is a garbage.
Proof:
By inconsistent system we can "prove" what ever we want with no limitations, but then our "proofs" are inconsistent.
A consistent system is based on a finite quantity of well-defined axioms, but then we can find in it statements which are well-defined by the consistent system but they cannot be proved by the current axioms of this system, and we need to add more axioms in order to prove these statements.
So any consistent system is limited by definition and any inconsistent system is not limited by definition.
Let us re-examine the universal quantification '
all'.
As I see it, when we use '
all' it means that everything is inside our domain and if our domain is infinitely many elements ,even if they are limited by some common property, the whole idea of "well-defined" domain of infinitely many elements, is an inconsistent idea.
For example:
Someone can say that [0,1] is an example of a well-defined domain, which is also a collection of infinitely many elements, but any examined transition from the internal collection of the infinitely many elements to 0 or 1, cannot be anything but a phase transition that terminates the state of infinitely many smeller states of the collection of the infinitely many elements, and we have in our hand a finite collection of different scales and 0 or 1.
In short, the well-defined ‘[0’ or ‘1]’ values and a collection of infinitely many elements that existing between them, has a XOR-like relations that prevents from us to keep the property of the internal collection as a collection of infinitely many elements, in an excluded-middle reasoning.
Again, it is clearly shown in:
http://www.geocities.com/complementarytheory/ed.pdf
Form this point of view a universal quantification can be related only to a collection of finitely many elements.
An example: LIM X---> 0, X*[1/X] = 1
In that case we have to distinguish between the word '
any' which is not equivalent here to the word '
all'.
'
any' is an inductive point of view on a collection of infinitely many elements, that does not try to capture everything by forcing a deductive '
all' point of view on a collection of infinitely many X values that cannot reach 0.
If my definition of '
all' is garbage, then 0*(1/0)=1
Since 0*(1/0) not= 1, Matt's response is a garbage. QED.