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A class of connected theories of order

Published online by Cambridge University Press:  12 March 2014

Alan S. Stern  and
Stanisław S. Świerczkowski
Alan S. Stern
Affiliation:
The Rowland Institute for Science, 100 Edwin H. Land Boulevard, Cambridge, Massachusetts 02142-1297, E-mail: [email protected]
Stanisław S. Świerczkowski
Affiliation:
Department of Mathematics and Computing, Sultan Qaboos University, P. O. Box 36 Al-Khod 123, Muscat, Sultanate of Oman
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Extract

In 1962 Jan Mycielski proposed a very general notion of interpretability [M1]. This led to the question whether a given theory could be interpreted in the disjoint union of two theories, without being interpretable in any of them. He argued that in such a case it would be presumably simpler to study each of these theories separately, and hence conjectured that this situation can never occur for any of the well-known theories of mathematics. This conjecture has now been verified for the following theories (see [MPS], [P], [S1, 2]): ELO (endless, i.e., without maximal element, linear order), Th(〈ℚ, ≤〉), Th(〈ω, ≤〉) and all sequential theories (those which can code finite sequences of elements of their models). The latter include PA, ZF, GB and Th(〈ω,+,·〉). In view of these confirmations it became ever more plausible that the conjecture is valid also for RCF (real closed fields), i.e., for Th(〈ℝ,≤,+,·,0,1〉). In the present paper we show that Mycielski's conjecture is valid for a class of theories which includes RCF and OF (ordered fields).

We consider only theories with equality and without function symbols. Interpretations will be meant local, multidimensional, and with parameters, as defined in [M1], [M2] and surveyed in [MPS] (for a recent definition see also [S2]). We shall write T0 ≪ T1 to say that T0 is interpretable in T1 (or that T1 interprets T0), and this will mean that for every theorem α of T0 there is an interpretation of α in T1.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 2 , June 1994 , pp. 534 - 542
Copyright
Copyright © Association for Symbolic Logic 1994

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References

REFERENCES

[M1] Mycielski, J., A lattice connected with relative interpretability of theories, Notices of the American Mathematical Society, vol. 9 (1962), pp. 407–408; erratum J. Mycielski, A lattice connected with relative interpretability of theories, Notices of the American Mathematical Society, vol. 18 (1971), p. 984.Google Scholar
[M2] Mycielski, J., A lattice of interpretability types of theories, this Journal, vol. 42 (1977), pp. 297–305.Google Scholar
[MPS] Mycielski, J., Pudlák, P. and Stern, A., A lattice of chapters of mathematics, Memoirs of the American Mathematical Society, vol. 86, American Mathematical Society, Providence, Rhode Island, 1991, pp. 1–70.Google Scholar
[P] Pudlák, P., Some prime elements in the lattice of interpretability types, Transactions of the American Mathematical Society, vol. 280 (1983), pp. 255–275.CrossRefGoogle Scholar
[S1] Stern, Alan S., The lattice of local interpretability of theories, Ph.D. Thesis , University of California, Berkeley, California, 1984.Google Scholar
[S2] Stern, Alan S., Sequential theories and infinite distributivity in the lattice of chapters, this Journal, vol. 54 (1989), pp. 190–206.Google Scholar