A cardinal $\kappa$ is $\Sigma\_n$-correct iff $V\_\kappa \prec\_n V$. For n>1, how to force a $\Sigma\_{n+1}$-correct cardinal to be $\Sigma\_{n}$-correct but not $\Sigma\_{n+1}$-correct? 


For $n=1$, we can force GCH below $\kappa$ and then violate GCH at $\kappa$.


If we assume some large cardinals, there are more partial answers, but the general situation is not clear to me.



How to kill a $\Sigma_{n+1}$-correct cardinal softly ($n>1$)?
This is a very interesting question!


My student Erin Carmody (PhD 2015) had asked this question in
connection with her dissertation, [Force to change large cardinal
strength](https://arxiv.org/abs/1506.03432), which contains many
such killing-them-softly results. Her theorem 20 is the
$\Sigma\_2$-reflecting cardinal case that you mention. I recall that
at the time we looked into the general $\Sigma\_n$-reflecting case,
but it was left open in her dissertation.


How nice finally to have an answer to this.


**Theorem.** If $\kappa$ is $\Sigma\_n$-correct, then there is a
class forcing extension preserving this in which $\kappa$ is not
$\Sigma\_{n+1}$-correct.


**Proof.** Suppose that $\kappa$ is $\Sigma\_n$ correct, where
$n>1$. By forcing, if necessary, let me assume that V=HOD holds and
more specifically that there is a $\Delta\_2$-definable
well-ordering of the universe. Indeed, let us arrange this
specifically by forcing to code every set into the GCH pattern on a
certain definable sequence of coding points. This forcing preserves
correctness.


Consider the forcing notion $\text{Add}(\kappa,1)$, which is
definable in $V\_\kappa$, and consider the dense sets for this
forcing that are $\Sigma\_n$-definable in $V\_\kappa$. I claim that
there is a subset $s\subset\kappa$ that is $\Sigma\_n$ definable in
$V\_\kappa$ and generic over $V\_\kappa$ with respect to
$\Sigma\_n$-definable dense classes for this forcing, meaning that
the initial segments of $s$ meet every $\Sigma\_n$-definable dense
subset of this forcing.


If $\kappa$ is inaccessible, then this is easier to see, since in
this case the forcing is $<\kappa$-closed and one can simply meet
the dense sets one by one, using a definable enumeration of them. A
careful version of this argument works even when $\kappa$ is
singular, using the fact that $\kappa$ is $\Sigma\_n$-correct. One
simply meets the dense sets one-by-one, and at limits, the
condition produced must be bounded below $\kappa$, since otherwise
one would have a $\Sigma\_n$-definable singularization of $\kappa$,
which would violate correctness.


Now let us force with $\text{Add}(\newcommand\Ord{\text{Ord}}\Ord,1)$ to add a generic class
$S\subset\Ord$ extending $s$. (We could also have used a definable
$\Sigma\_{n+1}$-generic class, without forcing.) In $V[S]$, let
$\mathbb{P}$ be the class forcing Easton support $\Ord$-iteration
that forces to code $S$ into the GCH pattern at another definable
sequence of ordinals that does not interfere with coding used
above.


Let $V[S][G]$ be the final extension. I claim that $\kappa$ remains
$\Sigma\_n$-correct, since if a $\Sigma\_n$-statement $\varphi(a)$ is
true in $V[S][G]$ about some parameter $a\in
V\_\kappa[s][G\_\kappa]$, then this is forced over $V[S][G]$ by some
condition, and by the correctness of $\kappa$ and the
$\Sigma\_n$-genericity of $s$, it follows that it holds in
$V\_\kappa[s][G\_\kappa]$ as desired. Basically, we are lifting the
relation $V\_\kappa\prec\_{\Sigma\_n} V$ to
$V[s][G\_\kappa]\prec\_{\Sigma\_n}V[S][G]$.


But I claim that $\kappa$ is not $\Sigma\_{n+1}$-correct in
$V[S][G]$. Note that $V$ is $\Sigma\_2$-definable in $V[S][G]$ using
the first sequence of coding points, and $V[S][G]$ thinks that $S$,
which is $\Sigma\_2$-definable, is not $\Sigma\_n$-definable in $V$,
since in fact it was generic over $V$. But $V\_\kappa[s][G\_\kappa]$
thinks that $s$ is $\Sigma\_n$-definable in $V\_\kappa$. This
violates $\Sigma\_{n+1}$-correctness. $\Box$


