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Extensional Independence

Taishi Kurahashi, Albert Visser

Abstract

Joel Hamkins asks whether there is a $Π^0_1$-formula $ρ(x)$ such that $ρ(φ)$ is independent over ${\sf PA}+φ$, if this theory is consistent, where this construction is extensional in $φ$ with respect to ${\sf PA}$-provable equivalence. We show that there can be no such extensional Rosser formula of any complexity. We give a positive answer to Hamkins' question for the case where we replace Extensionality by a weaker demand *Consistent Extensionality*. We also prove that we can demand the negation of $ρ$ to be $Π^0_1$-conservative, if we ask for the still weaker *Conditional Extensionality*. We show that an intensional version of the result for Conditional Extensionality cannot work.

Extensional Independence

Abstract

Joel Hamkins asks whether there is a -formula such that is independent over , if this theory is consistent, where this construction is extensional in with respect to -provable equivalence. We show that there can be no such extensional Rosser formula of any complexity. We give a positive answer to Hamkins' question for the case where we replace Extensionality by a weaker demand *Consistent Extensionality*. We also prove that we can demand the negation of to be -conservative, if we ask for the still weaker *Conditional Extensionality*. We show that an intensional version of the result for Conditional Extensionality cannot work.

Paper Structure

This paper contains 16 sections, 25 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. History of the Paper
  3. Provability predicates and related background
  4. There are no extensional Rosser Formulas
  5. There is a consistently extensional Rosser Formula
  6. A conditionally extensional strongly independent Formula
  7. Conventions and basic Definitions
  8. The $\Pi^0_1$-conservativity of Inconsistency
  9. Uniform Provability
  10. Uniform Smallness
  11. Uniform Slow Provability
  12. Density Revisited
  13. On intensionally finite Extensions
  14. Failure of Monotonicity Revisited
  15. Arrow Notation
  16. ...and 1 more sections

Key Result

Theorem 1.1

There is a $\Delta^0_2$-formula $\rho(x)$ over PA with the following properties.

Theorems & Definitions (56)

  • Theorem 1.1: Shavrukov and Visser shav:unif14
  • Corollary 1.2
  • Theorem 1.3: Shavrukov and Visser shav:unif14
  • Lemma 2.1: cf. Lindström lind:aspe03
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • ...and 46 more