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.