{Mathematical expression}free algebras

In this paper the study of which varieties, in a countable similarity type, have non-free {Mathematical expression} (or equivalently א₁-free) algebras is completed. It was previously known that if a variety satisfies a property known as the construction principle then there are such algebras. If a variety does not satisfy the construction principle then either every {Mathematical expression}-free algebra is free or for every infinite cardinal k, there is a k⁺-free algebra of cardinality k⁺ which is not free. Under the set theoretic assumption V=L, for any variety V in a countable similarity type, either the class of free algebras is definable in {Mathematical expression} or it is not definable in any {Mathematical expression}.