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A transfinite sequence of ω-models

Published online by Cambridge University Press:  12 March 2014

Andrzej Mostowski
Andrzej Mostowski*
Affiliation:
University of Warsaw, Warsaw, Poland
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Extract

Let A2 be the axiomatic system of second order arithmetic as described in [2]. One of the models of A2 is the “principal model” Mpr consisting of all integers and all sets of integers. Obviously there exist many denumerable ω-models elementarily equivalent to Mpr and we shall deal in this paper with some questions pertaining to this family which we denote by .

In §1 we define a rather natural relation ε between two denumerable families of sets of integers. From the upward Skolem-Löwenheim theorem it follows easily that there exists a family ordered by ε in the type ω1, but it is not immediately obvious whether there exist a subfamily of not well-ordered by ε. In the present paper we construct such a family of type η. ω1 where η is the order type of rationals and indicate some applications to hyperdegrees.

We adopt the terminology and notation of [2], with the only change that we adjoin to the language of A2 the constants ν0, ν1, … for the consecutive numerals 0, 1, 2, … and axioms which characterise them:

Also we modify the axioms of A2 given in [2] by prefixing them by general quantifiers bounded either to S or to N.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 37 , Issue 1 , March 1972 , pp. 96 - 102
Copyright
Copyright © Association for Symbolic Logic 1972

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References

REFERENCES

[1]Grzegorczyk, A., Mostowski, A. and Ryll-Nardzewski, Cz., The classical and the ω-complete arithmetic, this Journal, vol. 23 (1958), pp. 188–206.Google Scholar
[2]Mostowski, A. and Suzuki, Y., On ω-models which are not β-models, Fundamenta Mathematicae, vol. 65 (1969), pp. 83–93.CrossRefGoogle Scholar
[3]Shoenheld, J. R., Mathematical Logic, Addison-Wesley, Reading, Massachusetts, 1967.Google Scholar
[4]Keisler, H. J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.Google Scholar
[5]Thomason, S. K., The forcing method and the upper semilattice of hyperdegrees, Transactions of the American Mathematical Society, vol. 129 (1967), pp. 38–57.CrossRefGoogle Scholar