Re: existence of initial algebras

Some of what Paul Levy asked about --- building initial lagebras as sets of well-founded trees --- is in an old paper of mine, Words, free algebras, and coequalizers (Fund. Math. 117 (1983) 117--160), but even then the idea certainly wasn't new. That paper was concerned (partly) with what happens in varieties of algebras with infinitary operations --- so there are identities to be imposed on the terms. As a result, even the existence of free algebras with countable-ary operations depends on the axiom of choice. Andreas Blass