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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.

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    $\begingroup$ What you mean is that the sentence is neither upward nor downward absolute. $\endgroup$
    – Wojowu
    Commented Jun 28, 2019 at 15:52
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    $\begingroup$ You can't achieve what you want: large cardinals imply projective absoluteness, so in their presence you cannot change the truth value of projective statements. $\endgroup$
    – Andrés E. Caicedo
    Commented Jun 28, 2019 at 22:40
  • $\begingroup$ (Unless by "extend" you do not mean "by set forcing".) $\endgroup$
    – Andrés E. Caicedo
    Commented Jun 28, 2019 at 22:49
  • $\begingroup$ Ok, but isn't it the case that I can firstly collapse large cardinals and then procede as before? I don't know much about them, but I don't see obstacles for doing that. $\endgroup$
    – Jarek Swaczyna
    Commented Jun 29, 2019 at 7:26
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    $\begingroup$ Not if there are a proper class of them $\endgroup$
    – Andreas Lietz
    Commented Jun 29, 2019 at 22:00

1 Answer 1

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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.

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  • $\begingroup$ Thanks, that makes sense! But I will wait with closing this question, maybe some other examples will appear. $\endgroup$
    – Jarek Swaczyna
    Commented Jun 29, 2019 at 9:06

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