Neither provable nor disprovable theorem I wonder about a theorem which can be proven that this theorem is $neither\, provable$ nor $disprovable$ using any kind of mathematical knowledge.
Questions:
1- Is there any such theorem? or is it proposable?
2- If there exists such a theorem about a fact, how should we think about it? does it mean that this fact is independent of everything?
 A: Mathematics is only theories based on set of axioms. From them, you can derive truth value of formula in your theory. It is well know that for any theory with arithmetics, there are unprovable formula (see Goedel).
What does it mean ? Just that your unprovable formula can be added to your set and you obtain a new theory. You can add the negation of the formula and you obtain another theory.
Both theory can be seen as valid (as long as the first one is). There is a well known example in geometry, where you can't prove that given a point and a line (disjoint from the point), there exists only one parallel to the line through the point. 
If you add this axiom to the other axioms of geometry you obtain the euclidean geometry, but if you don't, you can obtain some hyperbolic geometry. Both are nice geometries with different domains of usefulness. So you can't say something is true independently of everything else, except for god, but it's not the point here ? 
A: Yes, there are many such “theorems”. These include well-known conjectures such as Continuum Hypothesis, and also many more obscure ones, like Whitehead problem. One might also include axiom of choice among those (well, assuming ZF is not inconsistent).
You might want to take a look at the list at Wikipedia.
That said, your question has nothing to do with statistics or probability theory, really.
A: This is a bit of a semantical issue.
Commonly the word theorem refers to a statement which is provable from a certain theory, for example "Zorn's lemma is a theorem of ZFC" is to say that we can prove Zorn's lemma is true from ZFC.
However sometimes the word "sticks" to a certain assertion (such as Zorn's lemma) and it then becomes a part of the name "Cantor's theorem"; "Hahn-Banach Theorem"; "Tychonoff's theorem"; and so on. For example, the Hahn-Banach theorem is not a theorem of ZF, but it is a theorem of ZFC.
Given a collection of axioms which is "nice" (in the sense that we can easily determine if an arbitrary statement is an axiom); as well strong enough to describe the natural numbers, then if this collection is consistent then there are statements which are true and it cannot prove nor disprove. This is The Incompleteness Theorem mentioned by others.
The Continuum Hypothesis is one example of this. There are models of ZFC in which the continuum hypothesis is true; other models of ZFC in which it is false. Similarly the axiom of choice cannot be proved from the axioms of ZF.
However, we are free to add to ZFC other axioms which are sufficient to prove that the continuum hypothesis is true; or we can add axioms which will prove the continuum hypothesis is false. Of course we cannot add both axioms, as that would generate an inconsistency in the system... and inconsistencies are bad.
For example from the axioms of ZFC+$\lozenge_{\omega_1}$ we can prove the continuum hypothesis is true. On the other hand from ZFC+PFA we can prove that the continuum hypothesis is false.
All in all when we say that $\varphi$ is independent from the theory $T$ it means that $T+\varphi$ is consistent (if $T$ was consistent to begin with, of course). In particular this means that $\varphi$ is not independent of the theory $T+\varphi$. This should answer your second question: no statement is independent from all theories.
A: To answer if something is proveable, you need to ask if a well-formed formula is provable from some theory (set of axiom). From the other answers, you can find the many example of these from set theory. However, there many examples of statements that you can not prove which most people are very familiar with.
For example, the commutativity axiom for groups is not proveable using the axioms of group theory. You would go about showing this unproveability by producing two models of the group axioms: one which satisfy the axiom of commutativity and one that does not. For example any commutative group (like $\mathbb{Z}$) and non-commutative group $S_3$. 
This method of producing models is essentially how you prove even the independence results in set theory. 
