Motivated by Tsirelson's implicitly defined pathological Banach space, T. Gowers asked whether explicitly defined Banach spaces must include an isomorphic copy of either c_0 or some l^p. J. Iovino and P. Casazza gave an affirmative answer for first order continuous logic and other compact logics. We extend their work to logics that are either countably compact or countably tight or k, greatly improving our previous work in this area. Noteworthy is our extensive use of the topology of function spaces (C_p-theory) and of double limit conditions. This is joint work with Clovis Hamel.

This is a rehearsal of a talk for an imminent AMS special session, so I will speak for 40 minutes and then take questions and give some proofs during the remaining time..