Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T18:42:55.733Z Has data issue: false hasContentIssue false
  • English
  • Français

The λ-calculus is ω-incomplete

Published online by Cambridge University Press:  12 March 2014

G. D. Plotkin
G. D. Plotkin*
Affiliation:
University of Edinburgh, Edinburgh EH8 9NN, Great Britain
Get access Rights & Permissions [Opens in a new window]

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 symbolsλ, ( , ) 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, (λx 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) = ∅.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 39 , Issue 2 , June 1974 , pp. 313 - 317
Copyright
Copyright © Association for Symbolic Logic 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

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[4] Jacopini, G., Il principio di estensionalita nell'assiomatica del λ-calcolo (unpublished).Google Scholar