Abstract
AbstractTheorems 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 Montalbán’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.
Publisher
Cambridge University Press (CUP)
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