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Categoricity regained

  • Erik Ellentuck (a1)

Extract

One of the earliest goals of modern logic was to characterize familiar mathematical structures up to isomorphism by means of properties expressed in a first order language. This hope was dashed by Skolem's discovery (cf. [6]) of a nonstandard model of first order arithmetic. A theory T such that any two of its models are isomorphic is called categorical. It is well known that if T has any infinite models then T is not categorical. We shall regain categoricity by

(i) enlarging our language so as to allow expressions of infinite length, and

(ii) enlarging our class of isomorphisms so as to allow isomorphisms existing in some Boolean valued extension of the universe of sets.

Let and be mathematical structures of the same similarity type where say R is binary on A. We write if f is an isomorphism of onto , and if there is an f such that . We say that P is a partial isomorphism of onto and write if P is a nonempty set of functions such that

(i) if fP then dom(f) is a substructure of , rng(/f) is a substructure of , and f is an isomorphism of its domain onto its range, and

(ii) if fP, aA, and bB then there exist g,hP, both extending f such that a ∈ dom(g) and b ∈ rng(h). Write if there is a P such that .

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References

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[1]Barwise, J., Back and forth thru infinitary logic, Studies in mathematics, vol. 8 (Model theory), Mathematical Association of America, Washington, D. C., 1973, pp. 534.
[2]Jech, T., Lectures in set theory, Springer-Verlag, Berlin and New York, 1971.
[3]Karp, C., Finite-quantifier equivalence, Theory of models, North-Holland, Amsterdam, 1965, pp. 407412.
[4]Ryll-Nardzewski, C., On theories categorical in power ℵ0, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 7 (1959), pp. 545548.
[5]Scott, D., Logic with denumerably long formulas and finite strings of quantifiers, Theory of models, North-Holland, Amsterdam, 1965, pp. 329341.
[6]Skolem, T., Peano's axioms and models of arithmetic, Mathematical interpretation of formal systems, North-Holland, Amsterdam, 1955, pp. 114.
[7]Takeuti, G. and Zaring, W. M., Axiomatic set theory, Springer-Verlag, Berlin and New York, 1970.

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