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(Extra)ordinary equivalences with the ascending/descending sequence principle

Marta Fiori-Carones, Alberto Marcone, Paul Shafer, Giovanni Soldà

Abstract

We analyze the axiomatic strength of the following theorem due to Rival and Sands in the style of reverse mathematics. "Every infinite partial order $P$ of finite width contains an infinite chain $C$ such that every element of $P$ is either comparable with no element of $C$ or with infinitely many elements of $C$." Our main results are the following. The Rival-Sands theorem for infinite partial orders of arbitrary finite width is equivalent to $\mathsf{I}Σ^0_2 + \mathsf{ADS}$ over $\mathsf{RCA}_0$. For each fixed $k \geq 3$, the Rival-Sands theorem for infinite partial orders of width $\leq\! k$ is equivalent to $\mathsf{ADS}$ over $\mathsf{RCA}_0$. The Rival-Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to $\mathsf{SADS}$ over $\mathsf{RCA}_0$. Here $\mathsf{RCA}_0$ denotes the recursive comprehension axiomatic system, $\mathsf{I}Σ^0_2$ denotes the $Σ^0_2$ induction scheme, $\mathsf{ADS}$ denotes the ascending/descending sequence principle, and $\mathsf{SADS}$ denotes the stable ascending/descending sequence principle. To our knowledge, these versions of the Rival-Sands theorem for partial orders are the first examples of theorems from the general mathematics literature whose strength is exactly characterized by $\mathsf{I}Σ^0_2 + \mathsf{ADS}$, by $\mathsf{ADS}$, and by $\mathsf{SADS}$. Furthermore, we give a new purely combinatorial result by extending the Rival-Sands theorem to infinite partial orders that do not have infinite antichains, and we show that this extension is equivalent to arithmetical comprehension over $\mathsf{RCA}_0$.

(Extra)ordinary equivalences with the ascending/descending sequence principle

Abstract

We analyze the axiomatic strength of the following theorem due to Rival and Sands in the style of reverse mathematics. "Every infinite partial order of finite width contains an infinite chain such that every element of is either comparable with no element of or with infinitely many elements of ." Our main results are the following. The Rival-Sands theorem for infinite partial orders of arbitrary finite width is equivalent to over . For each fixed , the Rival-Sands theorem for infinite partial orders of width is equivalent to over . The Rival-Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to over . Here denotes the recursive comprehension axiomatic system, denotes the induction scheme, denotes the ascending/descending sequence principle, and denotes the stable ascending/descending sequence principle. To our knowledge, these versions of the Rival-Sands theorem for partial orders are the first examples of theorems from the general mathematics literature whose strength is exactly characterized by , by , and by . Furthermore, we give a new purely combinatorial result by extending the Rival-Sands theorem to infinite partial orders that do not have infinite antichains, and we show that this extension is equivalent to arithmetical comprehension over .

Paper Structure

This paper contains 8 sections, 33 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reverse mathematics background
  3. From one principle to many
  4. First proofs of RSpo_(< infinity) and (0,cof)-RSpo_(< infinity)
  5. Proofs of RSpo_(< infinity), RSpo_k, and (0,cof)-RSpoCD_2 in RCA_0 + ISigma_2 + ADS and below
  6. Reversals and equivalences
  7. Extending the Rival--Sands theorem to partial orders without infinite antichains
  8. Maximal chain principles and comments on the original Rival--Sands proof

Key Result

Lemma 2.1

The following are equivalent over $\mathsf{RCA}_0$.

Figures (1)

  • Figure 1: Selected principles and systems and their implications and non-implications over $\mathsf{RCA}_0$. An arrow indicates that the source principle/system implies the target principle/system over $\mathsf{RCA}_0$. No further arrows may be added, except those that may be inferred via transitivity. No arrows reverse. Proofs of these implications and separations may be found in ChongLemppYangChongSlamanYangHirschfeldtShoreHirstThesisLermanSolomonTowsnerLiuRT22vsWKLPateyImmunitySeetapunSlamanSimpsonSOSOA.

Theorems & Definitions (82)

  • Lemma 2.1: SimpsonSOSOA*Lemma III.1.3
  • Theorem 2.2: MarconeNW*Theorem 6.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.5: essentially Kierstead KiersteadDilworth
  • Definition 3.6
  • Definition 3.7
  • Proposition 3.8
  • ...and 72 more