On the relation between choice and comprehension principles in second order arithmetic

Journal of Symbolic Logic 51 (2):360-373 (1986)
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Abstract

We give a new elementary proof of the comparison theorem relating $\sum^1_{n + 1}-\mathrm{AC}\uparrow$ and $\Pi^1_n -\mathrm{CA}\uparrow$ ; the proof does not use Skolem theories. By the same method we prove: a) $\sum^1_{n + 1}-\mathrm{DC} \uparrow \equiv (\Pi^1_n -CA)_{ , for suitable classes of sentences; b) $\sum^1_{n+1}-DC \uparrow$ proves the consistency of (Π 1 n -CA) ω k, for finite k, and hence is stronger than $\sum^1_{n+1}-AC \uparrow$ . a) and b) answer a question of Feferman and Sieg

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Andrea Cantini
Università degli Studi di Firenze

Citations of this work

Fixed point theories and dependent choice.Gerhard Jäger & Thomas Strahm - 2000 - Archive for Mathematical Logic 39 (7):493-508.
Understanding uniformity in Feferman's explicit mathematics.Thomas Glaß - 1995 - Annals of Pure and Applied Logic 75 (1-2):89-106.

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References found in this work

Theory of Recursive Functions and Effective Computability.Hartley Rogers - 1971 - Journal of Symbolic Logic 36 (1):141-146.
Recursion-Theoretic Hierarchies.Wayne Richter - 1983 - Journal of Symbolic Logic 48 (2):497-498.
Recursive Function Theory and Logic.Ann Yasuhara - 1975 - Journal of Symbolic Logic 40 (4):619-620.
Second order arithmetic and related topics.K. R. Apt & W. Marek - 1974 - Annals of Mathematical Logic 6 (3):177.

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