Continuous Fraisse Conjecture
  We investigate the relation of countable closed subsets of the reals with respect to continuous monotone embeddability; we show that there are exactly aleph_1 many equivalence classes with respect to this embeddability relation.   This is an extension of Laver's 1971 result, who considered (plain) embeddability, which yields coarser equivalence classes.   Using this result we show that there are only countably many different Godel logics. 
Arnold Beckmann, Martin Goldstern, Norbert Preining