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:
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$T$ 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 $G$.
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$T$ 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:
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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 $G$.
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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!
Best Answer
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, 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, 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.