Notre Dame Journal of Formal Logic

Embedding and Coding below a 1-Generic Degree

Noam Greenberg and Antonio Montalbán


We show that the theory of 𝒟(≤ g), where g is a 2-generic or a 1-generic degree below 0ʹ, interprets true first-order arithmetic. To this end we show that 1-genericity is sufficient to find the parameters needed to code a set of degrees using Slaman and Woodin's method of coding in Turing degrees. We also prove that any recursive lattice can be embedded below a 1-generic degree preserving top and bottom.

Article information

Notre Dame J. Formal Logic, Volume 44, Number 4 (2003), 200-216.

First available in Project Euclid: 29 July 2004

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Zentralblatt MATH identifier

Primary: 03D28: Other Turing degree structures 03D35: Undecidability and degrees of sets of sentences

generic degree elementary theory lattice embedding


Greenberg, Noam; Montalbán, Antonio. Embedding and Coding below a 1-Generic Degree. Notre Dame J. Formal Logic 44 (2003), no. 4, 200--216. doi:10.1305/ndjfl/1091122498.

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