Let $\mathcal{M}_A=\langle\mathbb N, 0^{\mathcal{M}_A}, s^{\mathcal{M}_A}, +^{\mathcal{M}_A}, \times^{\mathcal{M}_A}, <^{\mathcal{M}_A}\rangle$ be the standard model for the language of arithmetic $\mathcal L_A$. Define theory $\text{Th}(\mathcal{M}_A)=\{\alpha : \mathcal{M}_A\models\alpha\}$.
I have doubts about whether Peano's axioms $\mathsf{PA}$ satisfy $\text{Th}(\mathcal{M}_A)=\{\alpha: \mathsf{PA}\models\alpha\}$. I think the answer is no, according to Godel's first incompleteness theorem. In the linked page, I found the paragraph which confirms my answer:
According to Gödel's incompleteness theorems, the theory of PA (if consistent) is incomplete. Consequently, there are sentences of first-order logic (FOL) that are true in the standard model of PA but are not a consequence of the FOL axiomatization. Essential incompleteness already arises for theories with weaker axioms, such as Robinson arithmetic.
One thing I don't understand is that how it is possible for the theory of PA to be inconsistent? There are different sets of Peano's axioms whose theory is inconsistent?
What are examples of sentences $\alpha\in\text{Th}(\mathcal{M}_A)$ that satisfies $\mathsf{PA}\not\models\alpha$?