Journal of Symbolic Logic

Embedding jump upper semilattices into the Turing degrees

Antonio Montalbán

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Abstract

We prove that every countable jump upper semilattice can be embedded in 𝒟, where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and 𝒟 is the jusl of Turing degrees. As a corollary we get that the existential theory of 〈D,≤T,∨,’〉 is decidable. We also prove that this result is not true about jusls with 0, by proving that not every quantifier free 1-type of jusl with 0 is realized in 𝒟. On the other hand, we show that every quantifier free 1-type of jump partial ordering (jpo) with 0 is realized in 𝒟. Moreover, we show that if every quantifier free type, p(x1,…,xn), of jpo with 0, which contains the formula x1≤ 0(m)∧…∧xn≤ 0(m) for some m, is realized in 𝒟, then every quantifier free type of jpo with 0 is realized in 𝒟.

We also study the question of whether every jusl with the c.p.p. and size κ≤ 20 is embeddable in 𝒟. We show that for κ=20 the answer is no, and that for κ=ℵ1 it is independent of ZFC. (It is true if MA(κ) holds.)

Article information

Source
J. Symbolic Logic, Volume 68, Issue 3 (2003), 989-1014.

Dates
First available in Project Euclid: 17 July 2003

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1058448451

Digital Object Identifier
doi:10.2178/jsl/1058448451

Mathematical Reviews number (MathSciNet)
MR2004h:03089

Zentralblatt MATH identifier
1059.03038

Citation

Montalbán, Antonio. Embedding jump upper semilattices into the Turing degrees. J. Symbolic Logic 68 (2003), no. 3, 989--1014. doi:10.2178/jsl/1058448451. https://projecteuclid.org/euclid.jsl/1058448451


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References

  • U. Abraham and R.A. Shore Initial segments of the degrees of size $\aleph_1$, Israel Journal of Mathematics, vol. 53 (1986), pp. 1--51.
  • C.J. Ash and J. Knight Computable Structures and the Hyperarithmetical Hierarchy, Elsevier Science,2000.
  • R.G. Downey Computability theory and linear orderings, Handbook of Recursive Mathematics, vol. 2, North Holland,1998, pp. 823--976.
  • J. Harrison Recursive pseudo-well-orderings, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 526--543.
  • P.G. Hinman Jump traces with large gaps, Recursion theory and complexity (M. M. Arslanov and S. Lempp, editors),1999, pp. 71--80.
  • P.G. Hinman and T.A. Slaman Jump embeddings in the Turing degrees, Journal of Symbolic Logic, vol. 56 (1991), pp. 563--591.
  • W. Hodges Model Theory, Cambridge Univeristy Press, Cambridge,1993.
  • T. Jech Set Theory, third millennium ed., Springer,2003.
  • C.G. Jockusch Jr and T.A. Slaman On the $\Sigma_2$ theory of the upper semilattice of the Turing degrees, Journal of Symbolic Logic, vol. 58 (1993), pp. 193--204.
  • S.C. Kleene and E.L. Post The upper semi-lattice of the degrees of recursive unsolvability, Annals of Mathematics, vol. 59 (1954), pp. 379--407.
  • K. Kunen Set Theory, An Introduction to Independence Proofs, North Holland, Amsterdam,1980.
  • A.H. Lachlan Distributive initial segments of the degrees of unsolvability, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 457--472.
  • S. Lempp and M. Lerman The decidability of the existential theory of the poset of the recursively enumerable degrees with jump relations, Advances in Mathematics, vol. 120 (1996), pp. 1--142.
  • M. Lerman Degrees of Unsolvability, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo,1983.
  • G.E. Sacks On suborderings of degrees of recursive unsolvability, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 7 (1961), pp. 46--56.
  • R.A. Shore and T.A. Slaman The $\forall\exists$ theory of $D(\leq,\vee,')$ is undecidable, In preparation.
  • T.A. Slaman and W.H. Woodin Definability in degree structures, Monograph in preparation.