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Multiple genericity: a new transfinite hierarchy of genericity notions

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Abstract

We introduce a transfinite hierarchy of genericity notions stronger than 1-genericity and weaker than 2-genericity. There are many connections with Downey and Greenberg’s hierarchy of totally α-c.a. degrees [8]. We give several theorems concerning the strength required to compute multiply generic degrees, and show that some of the levels in the hierarchy can be separated, and that these separations can be witnessed by a \(\Delta _2^0\) degree. Finally, we consider downward density for these classes.

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Authors and Affiliations

  1. School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Michael McInerney & Keng Meng Ng

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  1. Michael McInerney
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  2. Keng Meng Ng
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Correspondence to Michael McInerney.

Additional information

Both authors are partially supported by MOE2015-T2-2-055.

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McInerney, M., Ng, K.M. Multiple genericity: a new transfinite hierarchy of genericity notions. Isr. J. Math. 250, 1–51 (2022). https://doi.org/10.1007/s11856-022-2331-5

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  • DOI: https://doi.org/10.1007/s11856-022-2331-5

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