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The consistency of a system of combinatory restricted generality*

Published online by Cambridge University Press:  12 March 2014

Haskell B. Curry
Haskell B. Curry*
Affiliation:
Pennsylvania State University, University Park, Pennsylvania 16802
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Extract

The system , proposed without a claim as to its consistency in [CLg, §15C], has turned out to entail a consequence which is extremely suspicious. For from the axiom ⊦LH it follows that

holds for arbitrary X; hence for X ≡ YH, where Y is a fixed-point combinator, we have

Whether or not this leads to an actual contradiction it would be interesting to know; but, no matter whether it does or not, (2) seems highly counterintuitive.

The aim of this note is to point out that a related system, here called , is demonstrably consistent and sufficient for all practical purposes served by , and at the same time to correct an error in [CLg, §15D2], which was discovered too late to be corrected on the proofs. This is formed from by dropping H from the list of θ's, and hence deleting ⊦LH and (1). The new system does not allow inferential rules to be converted into formulas with the same ease as does ; but if one is content with stating the results as rules, is adequate for all the main results deduced for in [CLg, §15C–D].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 3 , September 1973 , pp. 489 - 492
Copyright
Copyright © Association for Symbolic Logic 1973

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Footnotes

*

This paper was prepared when the author was Visiting Mellon Professor at the University of Pittsburgh.

References

BIBLIOGRAPHY

[STB]Bunder, Martin V., Set theory based on combinatory logic, Thesis, University of Amsterdam, 1969.Google Scholar
[CLg]Curry, Haskell B., et al., Combinatory logic. Vol. I (with Feys, Robert), North-Holland, Amsterdam, 1959; Vol. II (with J. Roger Hindley and Jonathan P. Seldin), North-Holland, Amsterdam, 1972.Google Scholar
[SIC]Seldin, Jonathan P., Studies in illative combinatory logic, Thesis, University of Amsterdam, 1968.Google Scholar