Let $\mathfrak{M}$ be a countable transitive model of set theory.

Let $L$ be some countable language and $M$ be a countable (in $M$) $L$-structure. My question is:

1.In $\mathfrak{M}$ can we carry the construction of Scott sentence of $M$ $\sigma(M)^\mathfrak{M}$?

2.. Is $\sigma(M)^\mathfrak{M}$ identical with the Scott sentence of $M$ $\sigma(M)$ in the real world?

Let $\mathfrak{M}$ be a countable transitive model of set theory.

Let $L$ be some countable language and $A$ be a countable (in $\mathfrak{M}$) $L$-structure. My question is:

1. In $\mathfrak{M}$ can we carry the construction of Scott sentence of $A$ $\sigma(A)^\mathfrak{M}$?

2. Is $\sigma(A)^\mathfrak{M}$ identical with the Scott sentence of $A$ in the real world?

Let $\mathfrak{M}$ be a countable transitive model of set theory.

Let $L$ be some countable language and $M$ be an $L$-structure. My question is:

1.In $\mathfrak{M}$ can we carry the construction of Scott sentence of $M$ $\sigma(M)^\mathfrak{M}$?

2.. Is $\sigma(M)^\mathfrak{M}$ identical with the Scott sentence of $M$ $\sigma(M)$ in the real world?

Let $\mathfrak{M}$ be a countable transitive model of set theory.

Let $L$ be some countable language and $M$ be a countable (in $M$) $L$-structure. My question is:

1.In $\mathfrak{M}$ can we carry the construction of Scott sentence of $M$ $\sigma(M)^\mathfrak{M}$?

2.. Is $\sigma(M)^\mathfrak{M}$ identical with the Scott sentence of $M$ $\sigma(M)$ in the real world?