{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T02:52:11Z","timestamp":1777517531627,"version":"3.51.4"},"reference-count":8,"publisher":"SAGE Publications","issue":"2","license":[{"start":{"date-parts":[[2023,6,21]],"date-time":"2023-06-21T00:00:00Z","timestamp":1687305600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Computability"],"published-print":{"date-parts":[[2023,6,21]]},"abstract":"<jats:p>We study a class of operators on Turing degrees arising naturally from ultrafilters. Suppose [Formula: see text] is a nonprincipal ultrafilter on \u03c9. We can then view a sequence of sets [Formula: see text] as an \u201capproximation\u201d of a set B produced by amalgamating the [Formula: see text] via [Formula: see text]: we set [Formula: see text]. This can be extended to the Turing degrees, by defining [Formula: see text]. The [Formula: see text] \u2013 which we call \u201cultrafilter jumps\u201d \u2013 resemble classical limit computability in certain ways. In particular, [Formula: see text] is always a Turing ideal containing [Formula: see text]. However, they are also closely tied to Scott sets: [Formula: see text] is always a Scott set containing [Formula: see text]. (This yields an alternate proof of the standard result in reverse mathematics that Weak Konig\u2019s Lemma is strictly weaker than arithmetic comprehension.)<\/jats:p>\n                  <jats:p>Our main result is that the converse also holds: if [Formula: see text] is a countable Scott set containing [Formula: see text], then there is some ultrafilter [Formula: see text] with [Formula: see text]. We then turn to the problem of controlling the action of an ultrafilter jump [Formula: see text] on two degrees simultaneously, and for example show that there are nontrivial degrees which are \u201clow\u201d for some ultrafilter jump. Finally, we study the structure on the set of ultrafilters arising from the construction [Formula: see text]; in particular, we introduce a natural preordering on this set and show that it is connected with the classical Rudin\u2013Keisler ordering of ultrafilters. We end by presenting two directions for further research.<\/jats:p>","DOI":"10.3233\/com-170176","type":"journal-article","created":{"date-parts":[[2023,6,24]],"date-time":"2023-06-24T03:04:02Z","timestamp":1687575842000},"page":"101-115","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["Limit computability and ultrafilters"],"prefix":"10.1177","volume":"12","author":[{"given":"Uri","family":"Andrews","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, 213 Van Vleck Hall, Madison, WI 53706, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingzhong","family":"Cai","sequence":"additional","affiliation":[{"name":"Hyperimmune Books, 620 Morning Creek Ln, Suwanee, GA 30024, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Diamondstone","sequence":"additional","affiliation":[{"name":"Google, 1600 Amphitheatre Pkwy, Mountain View, CA 94043, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Noah","family":"Schweber","sequence":"additional","affiliation":[{"name":"Proof School, 515 Highland Ave., San Mateo, CA 94401, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2023,6,21]]},"reference":[{"key":"ref001","unstructured":"W.W.\u00a0Comfort and S.\u00a0Negrepontis, The Theory of Ultrafilters, Die Grundlehren der mathematischen Wissenschaften, Vol.\u00a0211, Springer-Verlag, New York\u2013Heidelberg, 1974."},{"key":"ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-68441-3"},{"key":"ref003","unstructured":"T.\u00a0Jech, Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, The third millennium edition, revised and expanded."},{"key":"ref004","doi-asserted-by":"publisher","DOI":"10.1017\/9781316755723.012"},{"key":"ref005","unstructured":"R.W.\u00a0Robinson, The Inclusion Lattice and Degrees of Unsolvability of the Recursively Enumerable Sets. ProQuest LLC, Ann Arbor, MI, 1966. Thesis (Ph.D.)\u2013Cornell University."},{"key":"ref006","unstructured":"N.D.\u00a0Schweber, Interactions Between Computability Theory and Set Theory. ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)\u2013University of California, Berkeley."},{"key":"ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-59971-2"},{"key":"ref008","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"}],"container-title":["Computability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/COM-170176","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/COM-170176","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/COM-170176","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T16:02:56Z","timestamp":1777392176000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/COM-170176"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,21]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,6,21]]}},"alternative-id":["10.3233\/COM-170176"],"URL":"https:\/\/doi.org\/10.3233\/com-170176","relation":{},"ISSN":["2211-3568","2211-3576"],"issn-type":[{"value":"2211-3568","type":"print"},{"value":"2211-3576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,21]]}}}