A Sneak Preview of Proof Theory of Ordinals( Infinity in Philosophy and Mathematics)

@article{Arai2011ASP,
  title={A Sneak Preview of Proof Theory of Ordinals( Infinity in Philosophy and Mathematics)},
  author={Toshiyasu Arai},
  journal={Annals of the Japan Association for Philosophy of Science},
  year={2011},
  volume={20},
  pages={29-47}
}
  • T. Arai
  • Published 3 February 2011
  • Mathematics, Philosophy
  • Annals of the Japan Association for Philosophy of Science
This talk is a sneak preview of the project, 'proof theory for theories of ordinals'. Background, aims, survey and furture works on the project are given. Subsystems of second order arithmetic are embedded in recursively large ordinals and then the latter are analysed. We scarcely touch upon proof theoretical matters. 
Lifting up the proof theory to the countables : Zermelo-Fraenkel set theory
We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound of definable countable ordinals in the Zermelo-Fraenkel's set theory ZF.
The Use of Trustworthy Principles in a Revised Hilbert’s Program
After the failure of Hilbert’s original program due to Godel’s second incompleteness theorem, relativized Hilbert’s programs have been suggested. While most metamathematical investigations are
Implicit Dynamic Function Introduction and Ackermann-like FunctionTheory
TLDR
The paper is based on recollections of Grigori Mints completed by a survey of his research work in a scientific context and sometimes goes beyond the purely scientific aspects to show the atmosphere of these times.
LIFTING PROOF THEORY TO THE COUNTABLE ORDINALS: ZERMELO-FRAENKEL SET THEORY
  • T. Arai
  • Mathematics, Philosophy
    The Journal of Symbolic Logic
  • 2014
Abstract We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound on definable countable ordinals in Zermelo-Fraenkel set theory ZF.

References

SHOWING 1-10 OF 48 REFERENCES
Proof-theoretic analysis of KPM
TLDR
It is shown that a certain ordinal notation system is sufficient to measure the proof-theoretic strength of KPM, which involves a detour through an infinitary calculus RS(M), for which several cutelimination theorems are proved.
Proof Theory of Reflection
Lifting up the proof theory to the countables : Zermelo-Fraenkel set theory
We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound of definable countable ordinals in the Zermelo-Fraenkel's set theory ZF.
Fragments of Kripke-Platek set theory with infinity
In this paper we shall investigate fragments of Kripke–Platek set theory with Infinity which arise from the full theory by restricting Foundation to Πn Foundation, where n ≥ 2. The strength of such
An ordinal analysis of parameter free Π12-comprehension
TLDR
This paper is the second in a series of three culminating in an ordinal analysis of Π12-comprehension, where KPi is KP with an additional axiom stating that every set is contained in an admissible set.
Collapsing functions based on recursively large ordinals: A well-ordering proof for KPM
SummaryIt is shown how the strong ordinal notation systems that figure in proof theory and have been previously defined by employing large cardinals, can be developed directly on the basis of their
Recent Advances in Ordinal Analysis: Π1 2 — CA and Related Systems
  • M. Rathjen
  • Mathematics
    Bulletin of Symbolic Logic
  • 1995
TLDR
Recent success is reported in obtaining an ordinal analysis for the system of Π2 analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to Π1-formulae, giving hope for an ordinals analysis of Z2 in the foreseeable future.
Proof Theory: An Introduction
This book contains the somewhat extended lecture notes of an introductory course in proof theory. As for the choice about the parts to he presented the anthor opts for what he considers to be the
...
...