My question is of a logical nature and concerns what I perceive to be two different types of mathematical independence.

Suppose we have a (sufficiently strong) axiomatic theory $T$. Gödel's Incompleteness Theorems state that:

$T$ is not a complete theory. That is, there is a sentence (expressible in the language of the theory) which is true, but not provable in the theory. In what follows, I will refer to such a sentence as a Gödel sentence and denote it by $G$.

$T$ cannot prove its own consistency. That is, assuming that $T$ is consistent, $\not\vdash_T\mathrm{Con}(T)$.

For my question to make sense, I must lay out the following principle, which I take to be "self-evident" (by which I mean that I believe most people would endorse it):

*'If one is committed to a theory $T$, then one is also committed to $\mathrm{Con}(T)$.'*

In other words, suppose that I accept the axioms of $PA$ (for instance). That means that I am committed to $PA$, in the sense that I believe it to be true, and therefore consistent. As such, it would be incoherent for me to disbelieve $\mathrm{Con}(PA)$.

This situation gives rise to the following state of affairs:

On the one hand, there are statements which are independent from a theory $T$, but whose truth is nevertheless

**implied**by $T$, even though $T$ cannot**prove**them. This is the paradigm of the First Gödel Theorem (cited above) applied to arithmetic: in the context of $PA$, it says that there is a Gödel sentence $G$ which is not provable in $PA$, but that if $PA$ is consistent, then $G$ must nevertheless be true. Thus, if one is committed to $PA$, one is committed to $\mathrm{Con}(PA)$ (by the above principle) and therefore one is committed to the truth of $G$.On the other hand, there are statements which are independent from a theory $T$, and in addition, no judgment regarding their truth value may be inferred from $T$. This is the paradigm of Set Theory ($T=ZFC$) and the Continuum Hypothesis ($CH$). One's commitment to $ZFC$ does not imply anything about the truth of $CH$, since both $ZFC+CH$ and $ZFC+\neg CH$ are consistent. Note that this is different from the first case, in which $PA+\neg G$ is inconsistent.

In essence, I see a dichotomy between statements which are independent from a theory $T$ **and also** from $\mathrm{Con}(T)$, and those which are independent from $T$ **but nevertheless implied** by $\mathrm{Con}(T)$. I am tempted to say that there are two types of logical independence; is such a division valid, or would anyone care to contest it?

In case this is a very well-known issue, are there any other examples (aside from Gödel sentences) of statements which are independent from a theory but provable if one assumes consistency? In particular, I am wondering if there are any "natural" such questions. (Of course, the statement $\mathrm{Con}(T)$ is itself an example, albeit a trivial one.)

Thank you!