An ordinal analysis for theories of self-referential truth

作者：Leigh Graham Emil^*; Rathjen Michael

来源：Archive for Mathematical Logic, 2010, 49(2): 213-247.

DOI：10.1007/s00153-009-0170-2

• Theories of truth
• Relative consistency and interpretations
• Ordinal analysis
• Cut elimination

The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard (Ann Pure Appl Log 33:1-21, 1987). There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised the following questions. Which subsets of the Optional Axioms are consistent over the base theory? What are the proof-theoretic strengths of the consistent theories? The first question was answered completely by Friedman and Sheard; all subsets of the Optional Axioms were classified as either consistent or inconsistent giving rise to nine maximal consistent theories of truth. They also determined the proof-theoretic strength of two subsets of the Optional Axioms. The aim of this paper is to continue the work begun by Friedman and Sheard. We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to well-known theories ranging from PA to the theory of Sigma(1)(1) dependent choice.

• 出版日期2010-3