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An unsolved problem in the theory of constructive order types

Published online by Cambridge University Press:  12 March 2014

Alan G. Hamilton*
Affiliation:
St. Catherine's College, Oxford

Extract

In the forthcoming monograph of Crossley [1] the question is raised whether the implication 2 + A = A ⇒ 1 + A = A is true for constructive order types. In this paper a partial answer to this question is given, in that a counterexample is constructed, using, however, not the definition of constructive order type (C.O.T.) given in [1], but an earlier one, namely that in Crossley [2]. The difference is that in [1] only orderings which can be imbedded in a standard dense r.e. ordering R by a partial recursive function are considered. The linear ordering constructed in this paper can be shown not to be such. The problem remains open in the case of orderings imbeddable in R.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1969

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References

[1]Crossley, J. N., Constructive order types, Monograph to be published.Google Scholar
[2]Crossley, J. N., “Constructive order types. I,” in Formal systtms and recursive functions, edited by Crossley, J. N. and Dummett, M. A. E., North-Holland, Amsterdam, 1965, pp. 189264.CrossRefGoogle Scholar
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