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A problem in the theory of constructive order types 1
Published online by Cambridge University Press: 12 March 2014
Extract
J. N. Crossley [1] raised the question of whether the implication 2 + A = A ⇒ 1 + A = A is true for constructive order types (C.O.T.'s). Using an earlier definition of constructive order type, A. G. Hamilton [2] presented a counterexample. Hamilton left open the general question, however, since he pointed out that Crossley considers only orderings which can be embedded in a standard dense r.e. ordering by a partial recursive function, and that his counterexample fails to meet this requirement. We resolve the question by finding a C.O.T. A which meets Crossley's requirement and such that 2 + A = A but 1 + A ≠ A. At the suggestion of A. B. Manaster and A. G. Hamilton we easily extend this construction to show that for any n ≧ 2, there is a C.O.T. A such that n + A = A but m + A ≠ A for 0 < m < n. Hence, Theorem 3 of [2] and all of its corollaries hold with the new definition of C.O.T. The construction is not difficult and requires no priority argument. The techniques are similar to those developed in [3], but no outside results are needed here.
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- Copyright © Association for Symbolic Logic 1970
Footnotes
The first author obtained a similar solution in May, 1968. This was unknown to the second author until after he independently solved the problem which was incorrectly announced [2] as unsolved in December, 1968. The second author was supported by National Science Foundation Grant GP 8866. We are grateful to A. B. Manaster and the referee for corrections.