Notre Dame Journal of Formal Logic

Filters on Computable Posets

Abstract

We explore the problem of constructing maximal and unbounded filters on computable posets. We obtain both computability results and reverse mathematics results. A maximal filter is one that does not extend to a larger filter. We show that every computable poset has a \Delta^0_2 maximal filter, and there is a computable poset with no \Pi^0_1 or \Sigma^0_1 maximal filter. There is a computable poset on which every maximal filter is Turing complete. We obtain the reverse mathematics result that the principle "every countable poset has a maximal filter" is equivalent to ACA₀ over RCA₀. An unbounded filter is a filter which achieves each of its lower bounds in the poset. We show that every computable poset has a \Sigma^0_1 unbounded filter, and there is a computable poset with no \Pi^0_1 unbounded filter. We show that there is a computable poset on which every unbounded filter is Turing complete, and the principle "every countable poset has an unbounded filter" is equivalent to ACA₀ over RCA₀. We obtain additional reverse mathematics results related to extending arbitrary filters to unbounded filters and forming the upward closures of subsets of computable posets.

Article information

Source
Notre Dame J. Formal Logic Volume 47, Number 4 (2006), 479-485.

Dates
First available in Project Euclid: 9 January 2007

http://projecteuclid.org/euclid.ndjfl/1168352662

Digital Object Identifier
doi:10.1305/ndjfl/1168352662

Mathematical Reviews number (MathSciNet)
MR2272083

Zentralblatt MATH identifier
1128.03037

Subjects
Primary: 03D 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35]
Secondary: 06

Citation

Lempp, Steffen; Mummert, Carl. Filters on Computable Posets. Notre Dame J. Formal Logic 47 (2006), no. 4, 479--485. doi:10.1305/ndjfl/1168352662. http://projecteuclid.org/euclid.ndjfl/1168352662.

References

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