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:


1. $T$ 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 $G$.
2. $T$ 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!



Logically independent but true sentences
The division you see has to do with the level of conservativity of the theories in question. On the one hand, the theory ZFC + Con(ZFC) is not $\Pi^0\_1$-conservative over ZFC since Con(ZFC) is a $\Pi^0\_1$ sentence which is not provable from ZFC. On the other hand, CH is $\Pi^0\_1$-conservative over ZFC since ZFC + CH and ZFC prove exactly the same arithmetical facts. Indeed, by the [Shoenfield Absoluteness Theorem](http://en.wikipedia.org/wiki/Absoluteness#Shoenfield.27s_absoluteness_theorem), ZFC and ZFC + CH prove exactly the same $\Sigma^1\_2$ facts. In general, forcing arguments will not affect $\Sigma^1\_2$ facts and so any statement whose independence is proved by means of forcing will be $\Sigma^1\_2$-conservative over ZFC. By contrast, large cardinal hypotheses are not $\Pi^0\_1$-conservative over ZFC.


An example of a natural statement that is independent of PA but nevertheless true is the [Paris–Harrington Theorem](http://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem), which is equivalent to the 1-consistency of PA. In other words, the statement is equivalent to the statement that every PA-provable $\Sigma^0\_1$ sentence is true.


