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My question is: within a system $S$ where $S$ is any extension of $Q$ (Robinson's Arithmetic), and when $P(y)$ is a provability predicate for $S$, when is $(X \rightarrow P(X))$ provable?

By definition of a provability predicate, it is always the case that if $X$ is provable in $S$ then $P(X)$ is provable in $S$. But this, as I understand it, is not the same as $(X \rightarrow P(X))$ being provable in $S$. So when is this the case?

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One instance where this holds is when $P$ is $\Sigma_1$ and $S$ is a bit stronger than $Q$: a sufficiently strong (consistent, recursively axiomatizable) theory of arithmetic (PA is massive overkill) proves that it proves every true $\Sigma_1$ sentence. I don't recall if $Q$ has this property.

We can get a partial reversal under a correctness assumption on $S$. Suppose $T$ is an appropriately-strong theory which proves "$S$ is sound" *(this is really a scheme: $T$ proves $P(X)\rightarrow X$ for each $X$, where again $P$ is the provability predicate for $S$)*. Then for each $X$, $T$ proves "If $S$ proves $X\rightarrow P(X)$, then $X\iff P(X)$," and $P(X)$ is a $\Sigma_1$ sentence regardless of the complexity of $X$.

  • Note that $S$ itself doesn't prove its own soundness, and see Lob's theorem.

Another example occurs whenever $S$ proves its own inconsistency, which can happen even if $S$ is consistent (consider PA + $\neg$Con(PA)). In this case $S$ trivially proves $X\rightarrow P(X)$ for each $X$, since $S$ proves $P(X)$ for each $X$.

  • Note that this means that we can't get a general reversal to the situation above.

Meanwhile, when you write

this, as I understand it, is not the same as $(X\rightarrow P(X))$ being provable in $S$,

you're absolutely right. Let's assume PA is sound. We know that the $\Sigma_2$-theory of arithmetic is not recursively enumerable, but the set of PA-provable $\Sigma_2$-sentences is; thus, since by soundness there is no false $\Sigma_2$-sentence which is PA-provable, there must be a true $\Sigma_2$-sentence which is not PA-provable. Letting $X$ be such a sentence, we have (again, by soundness of PA) that PA can't prove $X\rightarrow P(X)$.

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  • $\begingroup$ Correct (+1), but everything you wrote about $\Sigma_2$ in the last paragraph is also correct for $\Pi_1$. And of course, we actually know some specific true but PA-unprovable $\Pi_1$ sentences, e.g., Con(PA) or Gödel's "I am PA-unprovable." $\endgroup$ – Andreas Blass Apr 2 at 18:03
  • $\begingroup$ Thank you very much for the detailed answer, very informative and an interesting read! $\endgroup$ – Atlas Apr 2 at 19:53

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