Abstract
Coping with infinite injury is a subtle and beautiful topic. The reader should study this short introductory chapter now, and then again after studying Chap. 11.
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Notes
- 1.
In the proof of the Friedberg-Muchnik Theorem, we did not label our requirements as P for positive and N for negative. Rather, each requirement, say \(R_{2e} : A\ne \Phi _e^B\), was both positive and negative. That is, \(R_{2e}\) sometimes needed to force a witness into A, and sometimes needed to keep small elements out of B. That’s why we used the neutral label R.
In the proof of the Sacks Splitting Theorem, requirement \(R_{e, i}\) certainly was negative. It could also be viewed as positive in the sense that it might put certain elements into \(B_{1-i}\) (but for the sole purpose of keeping them out of \(B_i\)).
- 2.
This is true for the first generation of priority tree arguments, sometimes called “double-jump” arguments. For the next generation, the daunting “triple-jump” arguments (which are not treated in this book), the relationship between level e and the requirements \(P_e\) and \(N_e\) can be more complicated.
- 3.
Arguably, it traces back to 1954 [De]. True stages are not only an algorithmic alternative to priority trees; in [CGS], for example, the concept of true stages is used in the proof of correctness of a complicated priority tree algorithm. Various generalizations of true stages are presented in [Mo].
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Supowit, K.J. (2023). Introduction to Infinite Injury. In: Algorithms for Constructing Computably Enumerable Sets. Computer Science Foundations and Applied Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-26904-2_10
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DOI: https://doi.org/10.1007/978-3-031-26904-2_10
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