Abstract
A maximal chain in a tree is called a path, and a tree is called bounded when all its paths contain leaves. This paper concerns itself with first-order theories of bounded trees. We establish some sufficient conditions for the existence of bounded end-extensions that are also partial elementary extensions of a given tree. As an application of tree boundedness, we obtain a conditional axiomatisation of the first-order theory of the class of trees whose paths are all isomorphic to some ordinal \(\alpha < \omega ^{\omega }\), given the first-order theories of certain classes of bounded trees.
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Notes
The implications \(\text {(1)} \Rightarrow \text {(2)} \Rightarrow \text {(3)}\) are immediate and the implication \(\text {(1)} \Rightarrow \text {(4)}\) is straightforward. The other implications can be proved similarly to the Tarski-Vaught criterion for elementary substructures: refer to [7, Proposition 4.31] for the proof of the elementary substructure analogue of \(\text {(3)} \Rightarrow \text {(1)}\); this proof can be modified to prove the implication \(\text {(4)} \Rightarrow \text {(1)}\) as well.
By the properties of characteristic formulas, a finite such set \(\left\{ a_i \right\} _{i \in I}\) exists.
Here the node \(\left( a,0\right) \in \left| {\mathfrak {T}} +_L {\mathfrak {F}}\right| \) is identified with the node \(a \in T\), and \(\left( b,1\right) \in \left| {\mathfrak {T}} +_L {\mathfrak {F}}\right| \) is identified with \(b \in F\), in order to keep the notation simple.
Again the nodes \(\left( a_n,0\right) ,\left( c,1\right) \in \left| {\mathfrak {T}} +_L {\mathfrak {F}}\right| \) are identified with the nodes \(a_n \in T\) and \(c \in F\) to keep the notation simple, with similar conventions elsewhere in the proof.
References
Backofen, R., Rogers, J., Vijay-Shankar, K.: A first-order axiomatization of the theory of finite trees. J. Log. Lang. Inform. 4(1), 5–39 (1995)
Doets, K.: Monadic \(\varPi _1^1\)-theories of \(\varPi _1^1\)-properties. Notre Dame J. Formal Logic 30(2), 224–240 (1989)
Doets, K.: Basic Model Theory. CSLI Publications & The European Association for Logic, Language and Information (1996)
Goranko, V.: Trees and finite branching. In: Proceedings of the 2nd Panhellenic Logic Symposium, pp. 91–101 (1999)
Goranko, V., Kellerman, R.: Approximating trees as coloured linear orders and complete axiomatisations of some classes of trees. To appear
Goranko, V., Kellerman, R.: Classes and theories of trees associated with a class of linear orders. Log. J. IGPL 19(1), 217–232 (2010)
Hedman, S.: A First Course in Logic. Oxford University Press, (2004)
Kellerman, R.: First-order aspects of tree paths. Log. J. IGPL 23(4), 688–704 (2015)
Rosenstein, J.: Linear Orderings. Academic Press, (1982)
Rothmaler, P.: Introduction to Model Theory. Gordon and Breach Science Publishers, (2000)
Schmerl, J.: On \(\aleph _0\)-categoricity and the theory of trees. Fund. Math. 94(2), 121–128 (1977)
Shelah, S.: The monadic theory of order. Ann. Math. 102(3), 379–419 (1975)
Acknowledgements
We thank an anonymous referee, along with Valentin Goranko, for their careful reading and helpful comments and suggestions on the paper.
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Kellerman, R. First-order theories of bounded trees. Arch. Math. Logic 61, 263–297 (2022). https://doi.org/10.1007/s00153-021-00789-0
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DOI: https://doi.org/10.1007/s00153-021-00789-0