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In all the eases eonsidered in this book, it turns out that one ean easily show the definable isomorphisms to be exaetly the ones that hold in every model of the ealeulus, and this faet establishes an important eonneetion between the identifieation of types that it is desirable to make from a programmer's point of view and the notion of isomorphisms of objeets in models of A-ealculus. 3 Let A and B be types. Then A ~ B {::::::} A~dB. Proof. ({=) trivial. (=}) Take the open term model of the ealculus.

This technique, unlike the semantic one, extends smoothly to the secondorder case, the only real difficulty being the characterization of invertible terms. p. 's defined as folIows. xVI ... zPI ... p. xXi then Po-(i) is Xi. p. x 2 ßTJ. Proof. By interpretation in the untyped calculus. 5. 9. ISOMORPHISMS AND THE LAMBDA CALCULUS 49 This allows to show com pleteness of T h2 for isomorphisms of ).. 2 ßTJ. 2 ßTJ7r*, though, such a simple syntactic characterization is not already available. Actually, in the previous chapter only a sub set of invertible terms is characterized, whieh is sufficient for the purposes of the completeness proof for Th~T· The guideline for the proof is to try to deal with the complexities risen by the different term constructors one at a time, in order to achieve a sort of factorization of the invertibility problem for the full calculus into the invertibility problem for a more manageable subclass of terms.

3: The theories of valid isomorphisms for ccc's. correspond directly to numerieal product and exponentiation. Using these facts, it is easy to show that two objects are isomorphie in the eategory 0/ finite sets if and only if they are equal as numerieal expressions. This means that we have reduced the problem of characterizing isomorphie types in the eategory 0/ finite sets to the problem of finding aIl valid numerieal equation between expressions built out of product, exponentiation and the constant 1 only.

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