Online research in philosophy

We present a list of open questions in reverse mathematics, including some relevant background information for each question. We also mention some of the areas of reverse mathematics that are starting to be developed and where interesting open question may be found.

Two linear orderings are equimorphic if each can be embedded into the other. We prove that every hyperarithmetic linear ordering is equimorphic to a recursive one. On the way to our main result we prove that a linear ordering has Hausdorff rank less than $\omega _{1}^{\mathit{CK}}$ if and only if it is equimorphic to a recursive one. As a corollary of our proof we prove that, given a recursive ordinal α, the partial ordering of equimorphism types of linear orderings of (...) Hausdorff rank at most α ordered by embeddablity is recursively presentable. (shrink)

We prove that every countable jump upper semilattice can be embedded in D, where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and D is the jusl of Turing degrees. As a corollary we get that the existential theory of $\langle D, \leq_{T}, \vee, '\rangle$ is decidable. We also prove that this result is not true about jusls with 0, by proving that not every quantifier free (...) 1-type of jusl with 0 is realized in D. On the other hand, we show that every quantifier free 1-type of jump partial ordering (jpo) with 0 is realized in D. Moreover, we show that if every quantifier free type, $p(x_{1},..., x_{n})$ , of jpo with 0, which contains the formula $x_{1} \leq 0{^(m)} \& ... \& x_{n} \leq 0^{(m)}$ for some m, is realized in D, then every quantifier free type of jpo with 0 is realized in D. We also study the question of whether every jusl with the c.p.p. and size $\kappa \leq 2^{\aleph 0}$ is embeddable in D. We show that for $\kappa = 2^{\aleph 0}$ the answer is no, and that for κ = ℵ1 it is independent of ZFC. (It is true if $MA(\kappa)$ holds.). (shrink)