Notre Dame Journal of Formal Logic

Self-Embeddings of Computable Trees

Stephen Binns, Bjørn Kjos-Hanssen, Manuel Lerman, James H. Schmerl, and Reed Solomon

Abstract

We divide the class of infinite computable trees into three types. For the first and second types, 0' computes a nontrivial self-embedding while for the third type 0'' computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up to isomorphism. We show that every infinite computable tree must have either an infinite computable chain or an infinite Π01 antichain. This result is optimal and has connections to the program of reverse mathematics.

Article information

Source
Notre Dame J. Formal Logic, Volume 49, Number 1 (2008), 1-37.

Dates
First available in Project Euclid: 6 January 2008

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1199649898

Digital Object Identifier
doi:10.1215/00294527-2007-003

Mathematical Reviews number (MathSciNet)
MR2376778

Zentralblatt MATH identifier
1204.03044

Subjects
Primary: 03B25: Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10]
Secondary: 03E30: Axiomatics of classical set theory and its fragments 03C62: Models of arithmetic and set theory [See also 03Hxx]

Keywords
quantifiers decidability hereditarily finite sets

Citation

Binns, Stephen; Kjos-Hanssen, Bjørn; Lerman, Manuel; Schmerl, James H.; Solomon, Reed. Self-Embeddings of Computable Trees. Notre Dame J. Formal Logic 49 (2008), no. 1, 1--37. doi:10.1215/00294527-2007-003. https://projecteuclid.org/euclid.ndjfl/1199649898


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References

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