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# The λ-calculus is ω-incomplete

## Extract

The ω-rule in the λ-calculus (or, more exactly, the λK-β, η calculus) is

In [1] it was shown that this rule is consistent with the other rules of the λ-calculus. We will show the rule cannot be derived from the other rules; that is, we will give closed terms M and N such that MZ = NZ can be proved without using the ω-rule, for each closed term Z, but M = N cannot be so proved. This strengthens a result in [4] and answers a question of Barendregt.

The language of the λ-calculus has an alphabet containing denumerably many variables a, b, c, … (which have a standard listing e 1, e 2, …), improper symbols λ, ( , ) and a single predicate symbol = for equality.

Terms are defined inductively by the following:

(1) A variable is a term.

(2) If M and N are terms, so is (MN); it is called a combination.

(3) If M is a term and x is a variable, (λ x M) is a term; it is called an abstraction.

We use ≡ for syntactic identity of terms.

If M and N are terms, M = N is a formula.

BV(M), the set of bound variables in M, and FV(M), its free variables, are defined inductively by

A term M is closed iff FV(M) = ∅.

## References

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[1] Barendregt, H. P., Combinatory logic and ω-rule, Fundamenta Mathematicae (to appear).
[2] Curry, H. B. and Feys, R., Combinatory logic. Vol. 1, North-Holland, Amsterdam, 1958.
[3] Curry, H. B., Hindley, J. R. and Seldin, J. P., Combinatory logic. Vol. 2, North-Holland, Amsterdam, 1972.
[4] Jacopini, G., Il principio di estensionalita nell'assiomatica del λ-calcolo (unpublished).

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