We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on (...) the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture. (shrink)

Let X be a separable Banach space and Q be a coanalytic subset of . We prove that the set of sequences in X which are weakly convergent to some eX and is a coanalytic subset of . The proof applies methods of effective descriptive set theory to Banach space theory. Using Silver’s Theorem [J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 60–64], this result leads to the following dichotomy theorem: if X is a Banach space, is (...) a regular method of summability and is a bounded sequence in X, then there exists a subsequence iL such that either there exists eX such that every subsequence iH of iL is weakly summable w.r.t. to e and QiH,e); or for every subsequence iH of iL and every eX with QiH,e)the sequence iH is not weakly summable to e w.r.t. . This is a version for weak convergence of an Erdös–Magidor result, see [P. Erdös, M. Magidor, A note on Regular Methods of Summability, Proc. Amer. Math. Soc. 59 232–234]. Both theorems obtain some considerable generalizations. (shrink)

We prove a recursion theoretic characterization of the Topological Vaught Conjecture in the Zermelo‐Fraenkel set theory by using tools from effective descriptive set theory and by revisiting the result of Miller that orbits in Polish G‐spaces are Borel sets.