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**Quasi finitely axiomatizable totally categorical theories.**
*(English)*
Zbl 0592.03018

G. Cherlin, L. Harrington and A. H. Lachlan have shown [ibid. 28, 103-135 (1985; Zbl 0566.03022)] that theories which are categorical in all infinite powers are not finitely axiomatizable. They conjectured that any such theory is quasi finitely axiomatizable (i.e. interdefinable with a theory in a finite language which is axiomatized by one axiom and the axiom schema of infinity).

The authors prove the conjecture for all almost strongly minimal theories. They show that quasi finite axiomatizability is preserved under biinterpretability. This allows to restrict the attention to standard models \({\mathfrak A}=(A,W)\), where W is a modular n-Grassmannian over which A is algebraic, and a O-definable projection partitions A into fibres indexed by elements from W all of the same fixed cardinality.

All modular n-Grassmannians are quasi finitely axiomatizable. The authors proceed in showing that the n-Grassmannians also possess so called nice enumerations. That means that for any set of initial segments of such an enumeration there exist only finitely many elements \(s_ i\), \(0<i<n\), so that for any other element from the set one of the \(s_ i's\) is conjugate to a subsegment with the same last element. It is then shown that \({\mathfrak A}\) is quasi finitely axiomatizable. The first step in that is to show that the language can be restricted to a finite subset. Then a quasi finite axiom system is given and its completeness proved. In both proofs the fact that W has a nice enumeration is the key property, since partial automorphisms are being extended along increasing initial segments.

The authors prove the conjecture for all almost strongly minimal theories. They show that quasi finite axiomatizability is preserved under biinterpretability. This allows to restrict the attention to standard models \({\mathfrak A}=(A,W)\), where W is a modular n-Grassmannian over which A is algebraic, and a O-definable projection partitions A into fibres indexed by elements from W all of the same fixed cardinality.

All modular n-Grassmannians are quasi finitely axiomatizable. The authors proceed in showing that the n-Grassmannians also possess so called nice enumerations. That means that for any set of initial segments of such an enumeration there exist only finitely many elements \(s_ i\), \(0<i<n\), so that for any other element from the set one of the \(s_ i's\) is conjugate to a subsegment with the same last element. It is then shown that \({\mathfrak A}\) is quasi finitely axiomatizable. The first step in that is to show that the language can be restricted to a finite subset. Then a quasi finite axiom system is given and its completeness proved. In both proofs the fact that W has a nice enumeration is the key property, since partial automorphisms are being extended along increasing initial segments.

### MSC:

03C35 | Categoricity and completeness of theories |

### Keywords:

categoricity; almost strongly minimal theories; quasi finite axiomatizability; biinterpretability; modular n-Grassmannians; nice enumeration### Citations:

Zbl 0566.03022
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\textit{G. Ahlbrandt} and \textit{M. Ziegler}, Ann. Pure Appl. Logic 30, 63--82 (1986; Zbl 0592.03018)

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### References:

[1] | Ahlbrandt, G., Totally categorical theories of modular type, Ph.D. Thesis (1984), Chicago |

[2] | Cherlin, G.; Harrington, L.; Lachlan, A. H., \(ℵ_0\)-categorical, \(ℵ_0\)-stable theories, Annals Pure Appl. Logic, 28, 103-135 (1985) · Zbl 0566.03022 |

[3] | Higman, G., Ordering by divisibility in abstract algebra, Proc. London Math. Soc., 2, 3, 326-336 (1952) · Zbl 0047.03402 |

[4] | T. Coquand, Unpublished result; T. Coquand, Unpublished result |

[5] | G. Martin, Ph.D. Thesis, New Brunswick (forthcoming).; G. Martin, Ph.D. Thesis, New Brunswick (forthcoming). |

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