We show that for every computable limit ordinal α, there is a
computable structure 𝒜 that is Δα⁰
categorical, but not relatively Δα⁰ categorical (equivalently, it
does not have a formally Σα⁰ Scott family). We also show
that for every computable limit ordinal α, there is a computable
structure 𝒜 with an additional relation R that is
intrinsically Σα⁰ on 𝒜, but not relatively
intrinsically Σα⁰ on 𝒜 (equivalently, it is not definable
by a computable Σα formula with finitely many parameters).
Earlier results in [7], [10], and [8] establish the
same facts for computable successor ordinals α.
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription.
Read more about accessing full-text
References
C. J. Ash, A construction for recursive linear orderings, Journal of Symbolic Logic, vol. 56 (1991), pp. 673--683.
C. J. Ash, C. G. Jockusch, Jr., and J. F. Knight, Jumps of orderings, Transactions of the American Mathematical Society, vol. 319 (1990), pp. 573--599.
Mathematical Reviews (MathSciNet):
MR955487
C. J. Ash and J. Knight, Computable structures and the hyperarithmetical hierarchy, Elsevier, Amsterdam, 2000.
C. J. Ash, J. Knight, M. Manasse, and T. Slaman, Generic copies of countable structures, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 195--205.
Mathematical Reviews (MathSciNet):
MR998606
S. A. Badaev, Computable enumerations of families of general recursive functions, Algebra and Logic, vol. 16 (1977), pp. 129--148 (Russian), 83--98 (English translation).
Mathematical Reviews (MathSciNet):
MR536674
J. Chisholm, Effective model theory vs. recursive model theory, Journal of Symbolic Logic, vol. 55 (1990), pp. 1168--1191.
S. S. Gončarov, The number of nonautoequivalent constructivizations, Algebra and Logic, vol. 16 (1977), pp. 257--282 (Russian), 169--185 (English translation).
Mathematical Reviews (MathSciNet):
MR516028
S. S. Goncharov, V. Harizanov, J. Knight, C. McCoy, R. Miller, and R. Solomon, Enumerations in computable structure theory, Annals of Pure and Applied Logic, vol. 136 (2005), pp. 219--246.
S. S. Goncharov, V. S. Harizanov, J. F. Knight, and R. A. Shore, $\Pi\sp 1\sb 1$ relations and paths through $\mathcalO$, Journal of Symbolic Logic, vol. 69 (2004), pp. 585--611.
M. S. Manasse, Techniques and counterexamples in almost categorical recursive model theory, Ph.D. thesis, University of Wisconsin, Madison, 1982.
H. Rogers, Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
Mathematical Reviews (MathSciNet):
MR224462
V. L. Selivanov, The numerations of families of general recursive functions, Algebra and Logic, vol. 15 (1976), pp. 205--226 (Russian), 128--141 (English translation).
Mathematical Reviews (MathSciNet):
MR476449
I. N. Soskov, Intrinsically hyperarithmetical sets, Mathematical Logic Quarterly, vol. 42 (1996), no. 4, pp. 469--480.
R. Watnick, A generalization of Tennenbaum's theorem on effectively finite recursive linear orderings, Journal of Symbolic Logic, vol. 49 (1984), pp. 563--569.
Mathematical Reviews (MathSciNet):
MR745385
