# Independence results about independence results

Define $Ind(ZFC,\sigma)$ to be the assertion "The sentence $\sigma$ is independent from ZFC".

I am looking for theorems in the form $Ind(ZFC, Ind (ZFC,\sigma))$.

Are there such theorems in set theory?

Of course the first order inexpressibility is a problem in these theorems and many of them are theorem schemas.

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Almost all instances of independence are instances of your further phenomenon, for the following general reason. Namely, if there is a model of ZFC which believes that "$\sigma$ is independent of ZFC" is true, then there is another model of ZFC which believes that it is false, since any assertion of independence fails in a model of ZFC plus $\neg$Con(ZFC), as there are no independent assertions for an inconsistent theory.
One might like to say that all instances of independence are instances of your further phenomenon, but unfortunately, we cannot quite prove that without further hypotheses. Indeed, it is consistent with ZFC that there are independent statements, but no instances of Ind(ZFC,Ind(ZFC,$\sigma$)). To see this, consider any model $M$ of ZFC+Con(ZFC)+$\neg$Con(ZFC+Con(ZFC)). Such a model exists by the incompleteness theorem, and in this model we'll have Con(ZFC) and hence all kinds of independent statements, but inside $M$ we will have no models of ZFC+Con(ZFC), and thus $M$ thinks that there are no models of Ind(ZFC,$\sigma$) for any $\sigma$, and hence $M$ thinks Ind(ZFC,Ind(ZFC,$\sigma$)) always fails.