Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 47, Number 4 (2006), 479-485.
Filters on Computable Posets
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.
Notre Dame J. Formal Logic, Volume 47, Number 4 (2006), 479-485.
First available in Project Euclid: 9 January 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 03D 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35]
Lempp, Steffen; Mummert, Carl. Filters on Computable Posets. Notre Dame J. Formal Logic 47 (2006), no. 4, 479--485. doi:10.1305/ndjfl/1168352662. https://projecteuclid.org/euclid.ndjfl/1168352662