Abstract
We show that there is an intermediate recursively enumerable Turing degree in every model of P −+BΣ1. The proof is not uniform, depending on whether IΣ1 holds. There is a model of P −+BΣ1 in which there is a least recursively enumerable degree strictly above the recursive degree. Thus, the Sacks Splitting Theorem cannot be proven in P −+BΣ1.
The first author was partially supported by a grant from the Japan Society for the Promotion of Science during the preparation of this paper. Both authors were partially supported by N.S.F. grants and by Presidential Young Investigator Awards.
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Slaman, T.A., Woodin, W.H. (1989). Σ1-Collection and the finite injury priority method. In: Shinoda, J., Tugué, T., Slaman, T.A. (eds) Mathematical Logic and Applications. Lecture Notes in Mathematics, vol 1388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083670
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DOI: https://doi.org/10.1007/BFb0083670
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