Name
Abstract | ||
|---|---|---|
Theorems of hyperarithmetic analysis (THAs) occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed (recursive) iterations of the Turing jump but below ATR(0) (and so Pi(1)(1)-CA(0) or the hyperjump). There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin's [9] work on rays in graphs. They seem to be typical applications of ACA(0) but are actually THAs. These results answer Question 30 of Montalban's Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity. This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis (ATHAs). When combined with ACA(0) they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes (Pi(1)(1), r-Pi(1)(1)], and Tanaka) are defined and these ATHAs as well as many other principles are shown to be conservative over RCA(0) in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA(0) but over ACA(0) is equivalent to the other which may be strong (THA) or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic. |
Year | DOI | Venue |
|---|---|---|
2022 | 10.1017/bsl.2021.70 | BULLETIN OF SYMBOLIC LOGIC |
Keywords | DocType | Volume |
hyperarithmetic analysis, conservative, Hahn type theorems, ubiquity | Journal | 28 |
Issue | ISSN | Citations |
1 | 1079-8986 | 0 |
PageRank | References | Authors |
0.34 | 0 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| James S. Barnes | 1 | 0 | 0.34 |
| Jun Le Goh | 2 | 0 | 0.68 |
| Richard A. Shore | 3 | 331 | 58.12 |