Abstract
The Hanf number for a set S of sentences in \(\mathcal {L}_{\omega _1,\omega }\) (or some other logic) is the least infinite cardinal \(\kappa \) such that for all \(\varphi \in S\), if \(\varphi \) has models in all infinite cardinalities less than \(\kappa \), then it has models of all infinite cardinalities. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is \(\beth _{\omega _1^{CK}}\). The same argument proves that \(\beth _{\omega _1^{CK}}\) is the Hanf number for Scott sentences of hyperarithmetical structures.
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References
Ash, C.J., Knight, J.F.: Computable Structures and the Hyperarithmetical Hierarchy. Elsevier, Amsterdam (2000)
Csima, B., Harizanov, V., Miller, R., Montalbán, A.: Computability of Fraïssé limits. J. Symb. Logic 76, 66–93 (2011)
Goncharov, S.S.: Strong constructivizability of homogeneous models. Algebra i Logika 17, 363–388 (1978)
Hodges, W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997)
Keisler, H.J.: Model Theory for Infinitary Logic. North-Holland, Amsterdam (1971)
Nadel, M.: \(L_{\omega _1,\omega }\) and admissible fragments. In: Barwise, K.J., Feferman, S. (eds.) Model-Theoretic Logics, pp. 271–316. Springer, New York (1985)
Nadel, M.: Scott sentences and admissible sets. Ann. Math. Logic 7, 267–294 (1974)
Peretyat’kin, M.G.: A criterion of strong constructivizability of a homogeneous model. Algebra i Logika 17, 436–454 (1978)
Ressayre, J.-P.: Boolean valued models and infinitary first order languages. Ann. Math. Logic 6, 41–92 (1973)
Ressayre, J.-P.: Models with compactness properties relative to an admissible language. Ann. Math. Logic 11, 31–55 (1977)
Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw Hill, New York (1967)
Scott, D.: Logic with denumerably long formulas and finite strings of quantifiers. In: Addison, J., Henkin, L., Tarski, A. (eds.) The Theory of Models, pp. 329–341. North-Holland, Amsterdam (1965)
Souldatos, I.: Characterizing the power set by a complete Scott sentence. Fundam. Math. 6, 131–154 (2013)
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Goncharov, S.S., Knight, J.F. & Souldatos, I. Hanf number for Scott sentences of computable structures. Arch. Math. Logic 57, 889–907 (2018). https://doi.org/10.1007/s00153-018-0615-6
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DOI: https://doi.org/10.1007/s00153-018-0615-6
Keywords
- Computable structures
- Scott sentences
- Infinitary Logic
- Hanf Number
- Characterizing cardinals
- Hyperarithmetical structures