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Omitting quantifier-free types in generic structures

Published online by Cambridge University Press:  12 March 2014

Angus Macintyre
Angus Macintyre*
Affiliation:
University of Aberdeen, Aberdeen, AB9 1AS, Scotland
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Extract

The central result of this paper was proved in order to settle a problem arising from B. H. Neumann's paper [10].

In [10] Neumann proved that if a finitely generated group H is recursively absolutely presentable then H is embeddable in all nontrivial algebraically-closed groups. Harry Simmons [14] clarified this by showing that a finitely generated group H is recursively absolutely presentable if and only if H can be recursively presented with solvable word-problem. Therefore, if a finitely generated group H can be recursively presented with solvable word-problem then H is embeddable in all nontrivial algebraically-closed groups.

The problem arises of characterizing those finitely generated groups which are embeddable in all nontrivial algebraically-closed groups. In this paper we prove, by a forcing argument, that if a finitely generated group H is embeddable in all non-trivial algebraically-closed groups then H can be recursively presented with solvable word-problem. Thus Neumann's result is sharp.

Our results are obtained by the method of forcing in model-theory, as developed in [1], [12]. Our method of proof has nothing to do with group-theory. We prove general results, Theorems 1 and 2 below, about constructing generic structures without certain isomorphism-types of finitely generated substructures. The formulation of these results requires the notion of Turing degree. As an application of the central result we prove Theorem 3 which gives information about the number of countable K-generic structures.

We gratefully acknowledge many helpful conversations with Harry Simmons.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 37 , Issue 3 , September 1972 , pp. 512 - 520
Copyright
Copyright © Association for Symbolic Logic 1972

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References

REFERENCES

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