Nonstandard Models in Recursion Theory and Reverse Mathematics

@article{Chong2014NonstandardMI,
  title={Nonstandard Models in Recursion Theory and Reverse Mathematics},
  author={Chi Tat Chong and Wei Li and Yue Yang},
  journal={Bull. Symb. Log.},
  year={2014},
  volume={20},
  pages={170-200}
}
We give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs. 
RAMSEY’S HEOREM FOR PAIRS IN REVERSE MATHEMATICS
  • Mathematics
  • 2020
We consider the combinatorial principleRT2 derived from Ramsey’s Theorem for pairs, and discuss its prooftheoretic strength within the framework of reverse mathematics. Some of the techniques
RAMSEY’S THEOREM FOR PAIRS: A CASE STUDY IN REVERSE MATHEMATICS
  • Mathematics
  • 2019
We consider the combinatorial principleRT2 derived from Ramsey’s Theorem for pairs, and discuss its prooftheoretic strength within the framework of reverse mathematics. Some of the techniques
OPEN QUESTIONS ABOUT RAMSEY-TYPE STATEMENTS IN REVERSE MATHEMATICS
TLDR
Questions which naturally arose during this study are state and the inability to answer those questions reveals some gaps in the understanding of the combinatorics of Ramsey’s theorem.
The reverse mathematics of Ramsey-type theorems
In this thesis, we investigate the computational content and the logical strength of Ramsey's theorem and its consequences. For this, we use the frameworks of reverse mathematics and of computable
On the strength of Ramsey's theorem for trees
Conservation theorems for the Cohesiveness Principle
We prove that the Cohesiveness Principle (COH ) is Π1 conservative over RCA0 + IΣn and over RCA0 + BΣn for all n ≥ 2. (The cases for n ≥ 3 are new.) Our approach is recursiontheoretic. We first
1-Generic Degrees Bounding Minimal Degrees Revisited
  • C. Chong
  • Computer Science
    Computability and Complexity
  • 2017
We show that over the base system \(P^-+\Sigma _2\)-bounding, the existence of a 1-generic degree \(<\mathbf {0''}\) bounding a minimal degree is equivalent to \(\Sigma _2\)-induction.
Comparing the strength of diagonally non-recursive functions in the absence of $\Sigma^0_2$ induction
We prove that the statement "there is a $k$ such that for every $f$ there is a $k$-bounded diagonally non-recursive function relative to $f$" does not imply weak K\"onig's lemma over $\mathrm{RCA}_0
Coloring trees in reverse mathematics
COMPARING THE STRENGTH OF DIAGONALLY NONRECURSIVE FUNCTIONS IN THE ABSENCE OF ${\rm{\Sigma }}_2^0$ INDUCTION
Abstract We prove that the statement “there is a k such that for every f there is a k-bounded diagonally nonrecursive function relative to f” does not imply weak König’s lemma over
...
...

References

SHOWING 1-10 OF 107 REFERENCES
Separating Principles below Ramsey's Theorem for Pairs
TLDR
This paper answers two open questions concerning subsystems below RT, specifically that ADS is not equivalent to CAC and that EM is not equivalents to RT.
Reverse mathematics and Ramsey's property for trees
TLDR
It is shown that there is a class of partial orderings for which Ramsey's Theorem for pairs is equivalent to ACA0, and Ramsey'sTheorem for singletons for the complete binary tree is stronger than .
Reverse mathematics, computability, and partitions of trees
TLDR
The reverse mathematics and computability theory of a form of Ramsey's theorem in which the linear n-tuples of a binary tree are colored is examined.
Friedberg Numbering in Fragments of Peano Arithmetic and α-Recursion Theory
  • Wei Li
  • Mathematics
    The Journal of Symbolic Logic
  • 2013
TLDR
It is proved that (1) over P − + BΣ2, the existence of a Friedberg numbering is equivalent to I΢2, and (2) for Lα, there is a Friedburg numbering if and only if the tame Σ2 projectum of α equals the Σ1 cofinality of α.
∑ n Definable Sets without ∑ n Induction
We prove that the Friedberg-Muchnik Theorem holds in all models of Σ 1 collection under the base theory P − +IΣ 0 . Generalizations to higher dimensional analogs are discussed. We also study the
THE METAMATHEMATICS OF STABLE RAMSEY'S THEOREM FOR PAIRS
We show that, over the base theory RCA0, Stable Ramsey's The- orem for Pairs implies neither Ramsey's Theorem for Pairs nor � 0-induction.
Higher recursion theory
Hyperarithmetic theory is the first step beyond classical recursion theory. It is the primary source of ideas and examples in higher recursion theory. It is also a crossroad for several areas of
Peano's Axioms and Models of Arithmetic
Subsystems of second order arithmetic
TLDR
The results show clear trends in the development of mathematics within Subsystems of Z2 and in particular in the areas of arithmetical comprehension and models of Sub system design.
...
...