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.
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