# Proving the existence of a $\Pi_1$-sentence in True Arithmetic that is independent of Peano Arithmetic

I am trying to wrap my head around how to prove the following statement:

There exists some $$\Pi_1$$-sentence $$A$$ such that $$A \in \textbf{TA}$$ but $$\{A, \neg A\} \cap \textbf{PA} = \emptyset$$.

$$\textbf{TA}$$ represents True Arithmetic which is the set of statements in the language of arithmetic that are true under the natural model $$\underline{\mathbb{N}}$$. $$\textbf{PA}$$ represents Peano Arithmetic which is the set of statements in the language of arithmetic that are logical consequences of the set of Peano axioms $$\Gamma$$. In this context, we are under the assumption that $$\textbf{PA}$$ is, in fact, sound.

Under the assumption of soundness of $$\textbf{PA}$$ and its axiomatizability we can begin to prove parts of this statement. If $$A \in \textbf{TA}$$ then we are sure that $$\neg A \notin \textbf{PA}$$ by the soundness of $$\textbf{PA}$$; moreover, since $$\textbf{PA}$$ is axiomatizable (hence $$\Gamma$$ is recursively enumerable) by the completeness of the first-order $$LK$$ proof system if $$A \in \textbf{PA}$$ then $$\textbf{PA} \vdash A$$. With this in mind, the main statement I am interested in proving might be the following:

There exists some $$\Pi_1$$-sentence $$A$$ such that $$A \in \textbf{TA}$$ but $$\textbf{PA} \not\vdash A$$.

Is this going to just be a simple usage of the Godel sentence $$G$$ or am I missing something?