Israel Journal of Mathematics

, Volume 178, Issue 1, pp 229–252

# The theory of the α degrees is undecidable

Article

## Abstract

Let α be an arbitrary Σ1-admissible ordinal. For each n, there is a formula φn($$\vec x,\vec y$$) such that for any relation R on a finite set F with n elements, there are α-degrees $${\vec p}$$ such that the relation defined by φn($$\vec x,\vec p$$) is isomorphic to R. Consequently, the theory of α-degrees is undecidable.

## Preview

### References

1. [1]
J. C. E. Dekker and J. Myhill, Retraceable sets, Canadian Journal of Mathematics 10 (1958), 357–373.
2. [2]
Sy. D. Friedman, Negative solutions to Post’s Problem II, Annals of mathematics 113 (1981), 25–43.
3. [3]
N. Greenberg, The role of true finiteness in the admissible recursively enumerable degrees, Memoirs of the American Mathematical Society 181(854) (2006), vi+99.Google Scholar
4. [4]
N. Greenberg and A. Montalbán, Embedding and coding below a 1-generic degree, Notre Dame Journal of Formal Logic 44(4) (2003), 200–216 (electronic) (2004).
5. [5]
N. Greenberg and R. A. Shore and T. A. Slaman, The theory of the metarecursively enumerable degrees, Journal of Mathematical Logic 6 (2006), 49–68.
6. [6]
R. B. Jensen, The fine structure of the constructible hierarchy, Annals of Pure and Applied Logic 4 (1972) 229–308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver
7. [7]
M. Lerman, On suborderings of the α-recursively enumerable α-degrees, Annals of Pure and Applied Logic 4 (1972), 369–392.
8. [8]
J. M. MacIntyre, Noninitial segments of the α-degrees, Journal of Symbolic Logic 38 (1973), 368–382.
9. [9]
G. E. Sacks and S. G. Simpson, The α-finite injury method, Annals of Pure and Applied Logic 4 (1972), 343–367.
10. [10]
G. E. Sacks, Higher Recursion Theory, Perspectives in Mathematical Logic, Springer-Verlag, Heidelberg, 1990.
11. [11]
R. A. Shore, Minimal α-degrees, Annals of Pure and Applied Logic 4 (1972), 393–414.
12. [12]
S. G. Simpson, First-order theory of the degrees of recursive unsolvability, Annals of Mathematics. Second Series 105 (1977), 121–139.
13. [13]
T. A. Slaman and W. H. Woodin, Definability in Degree Structures, preprint.Google Scholar
14. [14]
T. A. Slaman and W. H. Woodin, Definability in the Turing degrees, Illinois Journal of Mathematics 30 (1986), 320–334.
15. [15]
C. Spector, On degrees of recursive unsolvability, Annals of Mathematics. Second Series 64 (1956), 581–592.
16. [16]
B. A. Trakhtenbrot, The impossibility of an algorithm for the decision problem for finite domains, Rossiiskaya Akademiya Nauk. Doklady Akademii Nauk 70 (1950), 569–572.Google Scholar
17. [17]
R. L. Vaught, Sentences true in all constructive models, The Journal of Symbolic Logic 25 (1960) 39–58.