4

Definability, unification and matching

4.1 Undecidability of lambda-definability

**The finite standard models**

Recall that the full type structure over a set *X*, written * _{X}*, is defined in Definition 2.4.18 as follows:

Note that if *X* is finite then all the *X*(*A*) are finite. In that case we can represent each element of * _{X}* by a finite piece of data and hence (through Gödel numbering) by a natural number. For instance for

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