I'm reading The problem of induction by John Vickers. The entry is quite comprehensive and enjoyable. There is a thing which seems a bit hasty, though. Vickers, at some point, makes a distinction between the problem of finding a method for distinguishing reliable inductive habits and the problem of saying what the difference between reliable and unreliable inductive habits is. While the distinction is quite sensible, I'm not sure what to think about the example Vickers uses to clarify it (section 2):
The distinction can be illustrated in the parallel case of arithmetic. The by now classic incompleteness results of the last century show that the epistemological problem for first-order arithmetic is insoluble; that there can be no method, in a quite clear sense of that term, for distinguishing the truths from the falsehoods of first-order arithmetic. But the metaphysical problem for arithmetic has a clear and correct solution: the truths of first-order arithmetic are precisely the sentences that are true in all arithmetic models.
A few remarks come to mind:
- Prima facie, the incompleteness of first order arithmetic (let's fix our attention on first-order true arithmetic TA1) would show that the epistemological problem of finding a method of deciding whether an arithmetical claim is true is unsolvable only if the only way to find out whether an arithmetical sentence is true would have to employ a complete axiomatization of true arithmetical sentences.
- Another way to put his is that I'm not sure if incompleteness itself entails the lack of epistemological criterion; after all there are negation-incomplete and decidable theories (think of the set of theorems of classical propositional logic). Yes, negation-completeness entails decidability, but the converse doesn't hold.
- For instance, if second-order model-theoretic consequence relation were decidable, we would have a decision method for first-order arithmetical truth despite the incompleteness theorem. It would suffice to check if a first-order arithmetical claim is a model-theoretic consequence of the axioms of second-order Peano arithmetic (which, modulo second-order semantic consequence, is complete).
- Perhaps one could circumvent the above concern by saying that if there were any epistemological criterion for arithmetical truth, then the true arithmetic would be recursively axiomatizable. This, alas, isn't immediately obvious and is quite sensitive to what your background epistemology is. In a scenario where one can have a direct contact with God (assume there is one) and God correctly responds to any queries we have about arithmetical truth, we do have some epistemological criterion for arithmetical truth, but this fact alone tells us nothing about the axiomatizability of arithmetical truth.
- Now, the set of first-order arithmetical truths in fact is undecidable. But this follows rather from the Church-Turing thesis, the undefinability of truth and the arithmetical definability of all recursive sets. If the set of FO arithmetical truths were recursive, it would be definable. But Tarski's theorem already states that it isn't. So the set of FO arithmetical truths isn't recursive, so by the Church-Turing thesis it is not decidable.
- Still, there is a gap between the claim that the set of arithmetical truths is undecidable and the claim that the epistemological problem for first-order arithmetics is unsolvable. This follows only if every epistemologically respectable criterion of arithmetical truth has to be a decision method. While this seems quite likely to me, there is still some maneuvering space for all those people/crackpots who disagree.
- It seems there is more to epistemological concerns about arithmetic than just incompleteness. If the incompleteness theorem failed, this would show that there are complete, sufficiently strong and consistent arithmetical theories. This by itself doesn't give us a full-blown epistemological criterion for arithmetical truth, because we also need some epistemological account of why the axioms of a given complete theory are true (after all, there are complete and false theories).
- Another worry is that Vickers defines FO arithmetical truths as those claims that hold in "all arithmetic models". Presumably he means all models of first-order arithmetic. The problem with this definition is that there are non-standard FO arithmetical models. For instance, I like to think that the Godel sentence for Peano Arithmetic is true. Yet, there is a model in which it is false. So Vickers's definition would exclude the Godel sentence from arithmetical truths. Also, since there are models in which the negation of the Godel sentence for PA is false, it also wouldn't be included in the set of arithmetical truths on Vickers's definition, thus yielding the set of arithmetical truths incomplete.