Abstract

This dissertation contains a number of results in the metatheory of inductive methods. The first chapter contains three main results. The first result shows that Bayesian norms on inductive methods are not inductively restrictive for ideal agents. The second result shows that Bayesian norms are restrictive for computable agents. The third result shows that a preference for reliable methods over unreliable ones only conflicts with one Bayesian axiom of ideally rational preference, and that axiom has no intuitive force as a constraint on ideal rationality. ;The second chapter contains a theorem expressing necessary and sufficient conditions for the existence of an inductive method that solves a discovery problem "unstably". Unstable success allows the scientist to vacillate among some finite set of hypotheses or theories in the limit, so long as eventually only true hypotheses are conjectured. First the existence of inductive problems that are unsolvable stably, but solvable stably, is demonstrated. Next unstable identification is characterized. Finally a sufficient condition for super-unstable success is given, where super-unstable success does not require that vacillation occur among only a finite set. ;The third chapter examines the empirical assessibility of hypotheses concerning limiting frequencies and randomness. The first two results show that a particular type of limiting frequency hypothesis is refutable but not verifiable in the limit even if we assume reasonably strong background knowledge. The next section shows that randomness is not decidable in the limit over the set of o-convergent sequences. The third section contains a theorem expressing the equivalence of the existence of a generalized kind of NP-test of a property and decidability of the property in the limit. ;The last chapter exhibits a connection between inductive methodology and the foundations of set theory. A proof technique known as diagonalization involves the construction of a "Cartesian demon" that fools the scientist no matter what method the scientist uses to make his conjectures. Diagonalization is used ubiquitously in proofs of unsolvability of inductive problems. A pressing question that arises is the following: Is diagonalization complete? That is, is there a Cartesian demon for every unsolvable problem? The answer turns out to depend upon the axioms of set theory