DUALITY AND DEFINABILITY IN FIRST ORDER LOGIC SECTION 1 7

[To be sure, S is not monadic over Set (because of the largeness of the similarity type L ),

but, taking all L-algebras in the Grothendieck universe U+ , we get a category 5, which is

(obviously) monadic over SET , the category of Z/.-sets, and thus the factorization fact will

hold in S* . Note that, starting with an arrow in the subcategory S of S, , the construction of

the regular factorization will stay entirely within S.]

Let us hasten to point out that there is a completely elementary (even trivial) proof of the

Beth definability theorem for Boolean algebras; nevertheless, the above proof is suggestive of

what is to come later.