On the undefinability of pathological Banach spaces

Motivated by Tsirelson's implicitly defined pathological Banach space, T. Gowers asked whether explicitly defined Banach spaces must include either $c_0$ or some $\ell^p$. J. Iovino and P. Casazza gave an affirmative answer for first-order continuous logic. We greatly extend their work to logics with much weaker requirements than compactness on their type spaces. Noteworthy is our extensive use of the topology of function spaces ($C_p$-theory) as developed by Arhangel'skii, and our use of double limit conditions studied by H. K\"onig and N. Kuhn.