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Transactions of the American Mathematical Society

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Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees


Author: C. J. Ash
Journal: Trans. Amer. Math. Soc. 298 (1986), 497-514
MSC: Primary 03D30; Secondary 03C57, 03C75, 03D45
Erratum: Trans. Amer. Math. Soc. 310 (1988), 851.
MathSciNet review: 860377
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Abstract: We show that, under certain assumptions of recursiveness in $ \mathfrak{A}$, the recursive structure $ \mathfrak{A}$ is $ \Delta _\alpha ^0$-stable for $ \alpha < \omega _1^{CK}$ if and only if there is an enumeration of $ \mathfrak{A}$ using a $ \Sigma _\alpha ^0$ set of recursive $ {\Sigma _\alpha }$ infinitary formulae and finitely many parameters from $ \mathfrak{A}$. This extends the results of [1].

To do this, we first obtain results concerning $ \Delta _\alpha ^0$ paths in recursive labelling systems, also extending results of [1]. We show, more generally, that a path and a labelling can simultaneously be defined, when each node of the path is to be obtained by a $ \Delta _\alpha ^0$ function from the previous node and its label.


References [Enhancements On Off] (What's this?)

  • [1] C. J. Ash, Stability of recursive structures in arithmetical degrees, Ann. Pure Appl. Logic 32 (1986), no. 2, 113–135. MR 863330, 10.1016/0168-0072(86)90048-5
  • [2] -, Categoricity of recursive structures in hyperarithmetical degrees (in preparation).
  • [3] E. Barker, Intrinsically $ \Sigma _\alpha ^0$ relations (preprint).
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  • [5] S. S. Gončarov, The number of nonautoequivalent constructivizations, Algebra i Logika 16 (1977), no. 3, 257–282, 377 (Russian). MR 516028
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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0860377-7
Article copyright: © Copyright 1986 American Mathematical Society