Epimorphisms, definability and cardinalities

Abstrakt

A class K of similar algebras is called a prevariety if it is closed under subalgebras, direct products and isomorphic images. This amounts to the demand that K be axiomatized by someclass Ξ of implications &{αi = βi : i ∈ I} =⇒ α = β (I a possibly infinite set) [2]. The claim that we cannot always find a set to play the role of Ξ is consistent with the set theory NBG (including choice). Its negation (i.e., the claim that sets suffice) is consistent with NBG if huge cardinals exist. These facts were established by Adamek.