Notion of truth in logic I've been studying Gödel's incompleteness theorems and I am stuck with the Tarskian notion of truth. I 've searched all over the internet, in books, in relevant topics here , but I didn't find a satisfactory answer. Ok , let me state exactly what my problem is :
The semantic version of the first incompleteness theorem states that there are true sentences of arithmetic which cannot be proven by PA. It perfectly makes sense to me that there are sentences which cannot be proven or refuted by PA but what does it mean to say that there are true sentences that cannot be proven by PA. The immediate response is that these sentences are true in the standard interpretation of PA, namely in the standard model of the natural numbers. But what does this mean ? When is a sentence "true" in the standard model of the natural numbers ? I know that the second-order Peano arithmetic is categorical and thus a unique set ( up to isomorphism ) is a model of these axioms. Therefore it seems that every sentence has a definite truth value, either it is true or it is false, according to whether it holds for that unique model. But that seems to me completely wrong. Let's take Goldbach's conjecture for the sake of argument. Suppose that it is neither provable or disprovable. In what sense does this conjecture hold ? Someone could say : trivially because IN FACT every even number can be written as a sum of two primes. Maybe it's the Aristotelian logic that I reject. I simply cannot accept that there is an absolute truth besides what we can prove. 
My current view is that Tarski's definition of truth in a formal system simply presupposes that we magically have a metasystem where every question has simply a definite answer. Then we just translate every sentence into its relative meta-sentence and decide whether the first sentence is true or not. Maybe that's a naive view. An analytical exposition would be great.
 A: One contemporary reading of Tarski's truth definition is that it simply defines truth in a structure. So, from that point of view, you do need to start with a particular structure before you can talk about "truth".  I don't believe this was Tarski's original aim in the 1930s, but it's a reasonable approach to "truth" from the current viewpoint of the field. 
So, yes, when people talk about "true but unprovable statements", or about a "standard model", they are beginning with the viewpoint that there is a "standard model" to which we can refer. Once someone accepts that there is a standard model, the fact that it gives each sentence a unique truth value is just a matter of Tarski's definition of truth in a structure. 
However, the reason that we know that Gödel sentence of a consistent theory $T$ is "true" but "unprovable in $T$" is that we can prove the Gödel sentence in some metatheory. If we couldn't prove the Gödel sentence, we wouldn't know it is true.
In general, if you don't like to talk about "truth", you can replace claims about "truth" with claims about provability in the metatheory. This is a very standard procedure, and authors often don't bother to even mention it.  They may write as if truth is well determined, but this still allows people who worry about it to replace "true" with some other meaning. (Similarly, authors may use Platonistic terminology, but people with other tastes know how to re-interpret the language to suit their taste.)
In particular, the Gödel sentence of a theory $T$ is provable in a very weak metatheory - primitive recursive arithmetic (PRA)  will suffice - under the additional assumption of Con($T$). This fact is part of a "formalized incompleteness theorem", where "formalized" means that we pay attention to the particular metatheory necessary to prove the theorem, including its claims about truth.
So what we have formally is that, when $T$ satisfies the hypotheses of the incompleteness theorems and $G_T$ is its Gödel sentence, then
$$
\text{PRA} \vdash \text{Con}(T) \to G_T.
$$
If we recognize the axioms of PRA as "true" in some sense, and that Con($T$) is "true" in that sense if and only if $T$ is consistent, then that formal result can be interpreted as: if $T$ is consistent then $G_T$ is "true" in the same sense.
A: 
I simply cannot accept that there is an absolute truth besides what we can prove.

This is a perfectly "sound" point of view, shared by Intuitionism and Constructivism.
But it is simply not the point of view of Tarski.
See e.g.:

*

*Alfred Tarski, The Semantic Conception of Truth and The Foundations of Semantics (1944) :


The predicate "true" is sometimes used to refer to psychological phenomena
such as judgments or beliefs, sometimes to certain physical objects, namely, linguistic expressions and specifically sentences, and sometimes to certain ideal entities called "propositions." [...] For several reasons it appears most convenient to apply the term "true" to sentences, and we shall follow this course.


The word "true," like other words from our everyday language, is certainly not unambiguous. And it does not seem to me that the philosophers who have discussed this concept have helped to diminish its ambiguity. In works and discussions of philosophers we meet many different conceptions of truth and falsity, and we must indicate which conception will be the basis of our discussion.


We should like our definition to do justice to the intuitions which adhere
to the classical Aristotelian conception of truth - intuitions which find
their expression in the well-known words of Aristotle's Metaphysics:



To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, or of what is not that it is not, is true.



If we wished to adapt ourselves to modern philosophical terminology, we could perhaps express this conception by means of the familiar formula:



The truth of a sentence consists in its agreement with (or correspondence to) reality.



If, on the other hand, we should decide to extend the popular usage of the term "designate" by applying it not only to names, but also to sentences, and if we agreed to speak of the designata of sentences as "states of affairs," we could possibly use for the same purpose the following phrase:



A sentence is true if it designates an existing state of affairs.


