Selected Work.

I have selected the papers that I like the most in three different areas.

Up to equimorphism, hyperarithmetic is recursive.  To appear in the Journal of Symbolic Logic.

Abstract. 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. This extends a result of Spector [1955] that says that every hypearithmetic well-ordering is isomorphic to a computable one.
   On the way to our main result we prove that a linear ordering has Hausdorff rank less than omega-1-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 $\alpha$, the partial ordering of equimorphism types of linear orderings of Hausdorff rank at most $\alpha$ ordered by embeddablity, is recursively presentable.

Embedding Jump Upper Semilattices into the Turing Degrees. Journal of Symbolic Logic, 68 (3), 2003, 989-1014.

Abstract. 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 D(≤T, v,') 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≤ 0^(m) &...& x_n≤ 0^(m) for some m, is realized in D, then every 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 κ ≤ 2^ω is embeddable in D. We show that for κ = 2^ω the answer is no, and that for κ = ω_1 it is independent of ZFC. (It is true if MA(κ) holds.)
 

Equivalence between Fraïssé's conjecture and Jullien's theorem. To appear in the Annals of Pure and Applied Logic.

Abstract. We say that a linear ordering L is extendible if every partial ordering that does not embed L can be extended to a linear ordering which does not embed L either. Jullien's theorem is a complete classification of the countable extendible linear orderings. Fraïssé's conjecture, which is actually a theorem, is the statement that says that the class of countable linear ordering is well quasiordered under embeddablity. In this paper we study the strength of these two theorems from the viewpoint of Reverse Mathematics. As a result of our analysis we get that they are equivalent over the basic system of  RCA0 + Sigma-1-1-induction.

We also prove that Fraïssé's conjecture is equivalent, over RCA0, to two other interesting statements. One that says that the class of well founded labeled tree, with labels from {+,-}, with very a natural order relation, is well quasiordered. The other statement says that every linear ordering which does not contain a copy of the rationals is equimorphic to a finite sum of indecomposable linear orderings.

While studying the proof theoretic strength of Jullien's thorem, we prove the extendibility of many linear orderings, including $\omega^2$ and $\eta$, using just ATR0 + Sigma-1-1-induction.