TITLE: How could a statement be true without proof?
QUESTION [8 upvotes]: Godel`s incompleteness theorem states that there may exist true statements which have no proofs in a formal system of particular axioms. Here I have two questions;
1) How can we say that a statement is true without a proof?
2) What has the self reference to do with this? Godel sentence "G" can say that SUB(a,a, no prove) but could be this just arbitrarily judgement about non-provability of "a" because it may simply has a proof which is not yet revealed or discovered?

REPLY [3 votes]: 1) How can we say that a statement is true without a proof?

The main idea behind the standard definitions of truth used by most mathematicians and logicians that study mathematical logic, model theory and related topics is the one given by Tarski. As you can see at the link, he separated the concept of truth from the syntax of the language (for a good number of reasons, e.g., to not allow statements such as "this statement is false"), i.e., the truth concept is not given within the language but by the metalanguage. As the concept of proof is syntactical, at first you won't need any proved formula to be true or any true formula to be provable, but there's a property called soundness and it gives the following result: 
In a first order theory, if $\Sigma \vdash \varphi$ then $\Sigma\vDash \varphi$.
First, $\Sigma \vdash \varphi$ means that a set of formulas $\Sigma$ can prove the formula $\varphi$, i.e., you have $\Sigma$ and logical axioms (and, maybe, some aditional nonlogical axioms from the theory) as assumptions and are able to derive or deduce $\varphi$ using the rules of inference of your theory, so this is a syntactical process. Now, in my notation, $\Sigma \vDash \varphi$ means that all structures (which carry the concept of truth) that satisfy the formulas in $\Sigma$ also satisfy $\varphi$, in other words, if all formulas in $\Sigma$ are true in some structure $\mathfrak{M}$, then $\varphi$ must be true in $\mathfrak{M}$. As $\Sigma$ usually is a set of axioms that define a theory, if $\mathfrak{M}$ is a structure that satisfies $\Sigma$ (i.e., all formulas in $\Sigma$ are true in $\mathfrak{M}$) we say that $\mathfrak{M}$ is a model of $\Sigma$. In his doctoral dissertation , Gödel proved the converse, which is known as his Completeness Theorem:   
In a first order theory, if $\Sigma \vDash \varphi$ then $\Sigma\vdash \varphi$.
We say that a theory is complete in that sense (some authors call it semantically complete, to not make confusion with other type of completeness that is used in the Incompleteness Theorems, which they may call syntactical completeness) if this statement is valid (and it is, Gödel proved it). 
Now we can observe the fact that in some theories there are statements that can't be proved as well as their negation, you don't need to go too far to have some examples of this happening, if you have a system where you axioms $\Sigma$ are the Field Axioms you can't prove the formula $\varphi$ where it is $(\exists x)(x^2=-1)$ for the following reason: we can see (in a not rigorous manner) both $\mathbb{C}$ and $\mathbb{R}$ as models of $\Sigma$ and $\varphi$ is true in $\mathbb{C}$ (as you can take $x=i$) and false in $\mathbb{R}$, which means that not all models of $\Sigma$ satisfy $\varphi$, i.e., $\Sigma \nvDash \varphi$ and, by the soundness property, we have that $\Sigma \nvdash \varphi$ and, with a similar argument, you also have that $\Sigma \nvdash \neg \varphi$. 
Now that we know that there are some theories that have statements that you can't prove (as well as their negation), we call such systems incomplete (or, as I commented early, syntactically incomplete), what Gödel showed in his First Incompleteness Theorem was that any first order theory that satisfy some conditions will be incomplete, and that proved that, because the way mathematicians axiomatize the standard mathematical theories, we can't scape from the fact that there will always be statements such as $\varphi$ I showed early. In particular this will be valid for some theories that have a set of axioms $\Sigma$ with some properties and such that $\textbf{PA}\subset \Sigma$ where $\textbf{PA}$ is the set of Peano's Axioms.
On the other hand, mathematicians suppose that there's a standard model for arithmetics, i.e., a structure $\mathfrak{N}$ that satisfies $\textbf{PA}$ and every arithmetical statement, to be considered true, must be satisfied in that structure. Now I can finally give you an answer to your question: by the First Incompleteness Theorems, loosely speaking, any system of axioms $\Sigma$ (given some conditions) for the standard arithmetic will have statements $\varphi$ such that  $\Sigma \nvdash \varphi$ and  $\Sigma \nvdash \neg \varphi$, but $\mathfrak{N}$ satisfies $\varphi$ or $\neg \varphi$, that means that there will always be a true statement (a formula satisfied by $\mathfrak{N}$) that can't be proven by a given system of axioms $\Sigma$.

2) What has the self reference to do with this? Godel sentence "G"
  can say that SUB(a,a, no prove) but could be this just arbitrarily
  judgement about non-provability of "a" because it may simply has a
  proof which is not yet revealed or discovered?

The self-reference is just an argument used by Gödel in his original proof from which he was able to construct a sentence that is not provable by the system and will be true when you see it with the lens of the meta-theory. With all I said I hope you can comprehend that some statements in a given fixed theory are impossible to prove, it's not a matter of something not being revealed or discovered. 
For further reference and lots of details and technicalities that are missing in my answer, as well as aspects of the problem consisting on deciding if a statement is true or not, I can recommend the following textbooks:

Hinman, Peter G. Fundamentals of Mathematical Logic. Wellesley, MA: A.K. Peters, 2005. 
Dalen, Dirk. Logic and Structure. Fifith Edition. Berlin: Springer-Verlag, 2013
Tourlakis, George. Lectures in Logic and Set Theory. Vol. 1, Mathematical Logic (Cambridge Studies in Advanced Mathematics ; 82). N.p.: Cambridge UP, 2003. 
Shoenfield, Joseph R. Mathematical Logic. Reading, MA: Addison-Wesley Pub., 1967. 

There are also good references and links here and here.