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Another infinite natural sum

Paolo Lipparini

Abstract

We define and study an infinitary operation on countable sequences of ordinals which is strictly monotone in many significant cases and has both an order-theoretical characterization and a characterization in terms of combinatorial games.

Another infinite natural sum

Abstract

We define and study an infinitary operation on countable sequences of ordinals which is strictly monotone in many significant cases and has both an order-theoretical characterization and a characterization in terms of combinatorial games.
Paper Structure (8 sections, 12 theorems, 30 equations)

This paper contains 8 sections, 12 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. The finitary natural sum and some obstacles to infinitary generalizations
  3. Some preliminary definitions and a lemma
  4. A strictly monotone operation
  5. An order-theoretical characterization
  6. A game theoretical characterization
  7. Further Remarks
  8. Appendix

Key Result

Proposition 2.3

CarT The natural sum $\mathbin{\hash}$ is the smallest binary operation on the ordinals which is strictly monotone on both arguments.

Theorems & Definitions (44)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 34 more