# What does it mean for two statements in a first-order language to be "independent" of each other?

More precisely, let $T$ be a theory of some first-order language. My question is then the following. What does it mean for two sentences $\sigma_1$ and $\sigma_2$ in the same language to be independent of each other in the theory $T$?

My "hypothesis" is that it means $\neg \, ( \, T \vdash \sigma_1 \leftrightarrow \sigma_2 \,)$. However, I could only come up with this "hypothesis" by looking at some particular examples; also I could not immediately find an online reference where this concept is defined, hence this question. Can somebody confirm this?

Note. I do know what it means for one formula $\sigma$ to be independent from a theory $T$. Namely it means that $\neg \, ( \, T \vdash \sigma\,) \; \wedge \; \neg \, ( T \vdash \neg \sigma)$.

It’s even stronger than that: it means that $T\cup\{\sigma_1\}\not\vdash\sigma_2$ and $T\cup\{\sigma_2\}\not\vdash\sigma_1$. In other words, given $T$, neither of $\sigma_1$ and $\sigma_2$ entails the other. For example, in the context of $\mathsf{ZF}$ set theory the continuum hypothesis and the axiom of choice are independent of each other: assuming that $\mathsf{ZF}$ is consistent, so are $\mathsf{ZF}+\mathsf{CH}+\neg\mathsf{AC}$ and $\mathsf{ZF}+\mathsf{AC}+\neg\mathsf{CH}$.
• Brian's answer is correct. But there is abuse, as you see. In this particular case, what Jech means is that $\mathsf{ZFA}+$the ordering principle\$ does not imply the axiom of choice. Sep 8, 2013 at 2:14