  Recent results show that a constraint satisfaction problem (CSP) defined over
rational numbers with their natural ordering has a solution if and only if it
has a definable solution. The proof uses advanced results from topology and
modern model theory. The aim of this paper is threefold. (1) We give a simple
purely-logical proof of the claim and show that the advanced results from
topology and model theory are not needed; (2) we introduce an intrinsic
characterisation of the statement "definable CSP has a solution iff it has a
definable solution" and investigate it in general intuitionistic set theories
(3) we show that the results from modern model theory are indeed needed, but
for the implication reversed: we prove that "definable CSP has a solution iff
it has a definable solution" holds over a countable structure if and only if
the automorphism group of the structure is extremely amenable.
