Does mathematics need new axioms? was the second of three plenary panel discussions held at the ASL annual meeting, ASL 2000, in Urbana-Champaign, in June, 2000. Each panelist in turn presented brief opening remarks, followed by a second round for responding to what the others had said; the session concluded with a lively discussion from the floor. The four articles collected here represent reworked and expanded versions of the first two parts of those proceedings, presented in the same order as the speakers appeared at the original panel discussion: Solomon Feferman (pp. 401–413), Penelope Maddy (pp. 413–422), John Steel (pp. 422–433), and Harvey Friedman (pp. 434–446). The work of each author is printed separately, with separate references, but the portions consisting of comments on and replies to others are clearly marked.

Lyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of first-order logic, there is an interpolant in which each relation symbol appears positively (negatively) only if it appears positively (negatively) in both the antecedent and the succedent of the given implication. We prove a similar, more general interpolation result with the additional requirement that, for some fixed tuple of unary predicates U, all formulae under consideration have all quantifiers explicitly relativised to one of the U. Under this stipulation, existential (universal) quantification over U contributes a positive (negative) occurrence of U.

It is shown how this single new interpolation theorem, obtained by a canonical and rather elementary model theoretic proof, unifies a number of related results: the classical characterisation theorems concerning extensions (substructures) with those concerning monotonicity, as well as a many-sorted interpolation theorem focusing on positive vs. negative occurrences of predicates and on existentially vs. universally quantified sorts.