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The inconsistency of

Published online by Cambridge University Press:  12 March 2014

M. W. Bunder
M. W. Bunder*
Affiliation:
University of Wollongong, Wollongong, N.S.W. 2500, Australia
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Extract

In [4] Curry raised the possibility that his system proposed in ξ15C of [3] might be inconsistent. In this paper this inconsistency is proved using a method also employed in [1].

From Curry's axiom ⊦LH, it follows that

holds for arbitrary X.

The other results from that are required are

Modus Ponens, and the Deduction Theorem for implication:

Assuming ⊦HA, we define as in [1]:

and let

where Y is the paradoxical (or fixed point) combinator.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 2 , June 1976 , pp. 467 - 468
Copyright
Copyright © Association for Symbolic Logic 1976

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References

REFERENCES

[1] Bunder, M. W., A paradox in illative combinatory logic, Notre Dame Journal of Formal Logic, vol. 11 (1970), pp. 467–470.CrossRefGoogle Scholar
[2] Bunder, M. W., Some inconsistencies in illative combinatory logic, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 20 (1974), pp. 71–73.CrossRefGoogle Scholar
[3] Curry, H. B., Hindley, J. R. and Seldin, J. P., Combinatory logic, Vol. II, North-Holland, Amsterdam, 1972.Google Scholar
[4] Curry, H. B., The consistency of a system of combinatory restricted generality, this Journal, vol. 38 (1973), pp. 489–492.Google Scholar
[5] Seldin, J. P., Studies in illative combinatory logic. Thesis, Amsterdam, 1968.Google Scholar
[6] Seldin, J. P., The Q-consistency of (unpublished).Google Scholar