TITLE: Examples of independent $\Sigma_4^1$ statements
QUESTION [5 upvotes]: As in the title, I'm looking for examples of $\Sigma^1_4$ (preferably complete) sentences which are independent from ZFC in both ways, namely given a model $V$ we can extend it to $V'$ where such a sentence holds, but also extend it to a model $V''$ where such sentence fails. 
By Shoenfield Theorem $\Sigma_4^1$ (or $\Pi^1_4$) is the lowest available complexity of such a formula, and that's why I'm looking for such examples.

REPLY [7 votes]: As a starting point, think about the sentence "There is a nonconstructible real." This is $\Sigma^1_3$ and clearly not downwards-absolute. However, it is upwards-absolute.
To get the desired situation, we "relativize" and consider the sentence 

There is some real $r$ such that every real $s$ is constructible relative to $r$.

(That is, for some $r\in\mathbb{R}$ we have $\mathbb{R}=\mathbb{R}\cap L[r]$.)
This is $\Sigma^1_4$, and is neither downwards nor upwards absolute.