Introduction & plan of the talk Reverse Mathematics is an - PowerPoint PPT Presentation

R EVERSE M ATHEMATICS OF SOME P RINCIPLES RELATED TO P ARTIAL O RDERS Giovanni Sold` a, University of Leeds Joint work with Marta Fiori Carones, Alberto Marcone and Paul Shafer April 25 th 2019

Introduction & plan of the talk Reverse Mathematics is an ongoing project whose goal is to measure the “logical strength” of the theorems of ordinary mathematics: the classification proceeds by analyzing the set existence axioms required to prove the theorems. As we will see during the talk, the typical “reverse mathematical” process goes as follows: given a weak base theory A and a theorem T (formalized in Second Order Arithmetic), we look for a set existence axiom S such that A ⊢ T ↔ S . This talk consists of two parts: in the first one, we introduce more formally Reverse Mathematics, Z 2 and its main subsystems. A standard reference is Simpson, 2010; in the second one, we study, in this perspective, a theorem about the combinatorics of infinite posets due to Rival and Sands (Rival and Sands, 1980). April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 2 / 18

Introduction & plan of the talk Reverse Mathematics is an ongoing project whose goal is to measure the “logical strength” of the theorems of ordinary mathematics: the classification proceeds by analyzing the set existence axioms required to prove the theorems. As we will see during the talk, the typical “reverse mathematical” process goes as follows: given a weak base theory A and a theorem T (formalized in Second Order Arithmetic), we look for a set existence axiom S such that A ⊢ T ↔ S . This talk consists of two parts: in the first one, we introduce more formally Reverse Mathematics, Z 2 and its main subsystems. A standard reference is Simpson, 2010; in the second one, we study, in this perspective, a theorem about the combinatorics of infinite posets due to Rival and Sands (Rival and Sands, 1980). April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 2 / 18

Reverse Mathematics Reverse Mathematics 1 The principle RS - po 2 Background on RS - po RS - po in Reverse Mathematics April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 3 / 18

Reverse Mathematics L 2 and Z 2 We will use the two-sorted ( numbers and sets of numbers ) language L 2 = { 0 , 1 , + , ⋅ , < , ∈ , =} . Z 2 is the theory whose axioms are: Peano Axioms, induction ( 0 ∈ X ∧ ∀ n ( n ∈ X → n + 1 ∈ X )) → ∀ n ( n ∈ X ) and the Comprehension Scheme ∃ X ∀ n ( n ∈ X ↔ ϕ ( n )) , with X not occurring free in ϕ . As we mentioned, in Reverse Mathematics we work with subsystems of Z 2 . Five of them turn out to be particularly important, and are called the Big Five. During this talk, we will mostly be concerned with the first three of them. April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 4 / 18

Reverse Mathematics L 2 and Z 2 We will use the two-sorted ( numbers and sets of numbers ) language L 2 = { 0 , 1 , + , ⋅ , < , ∈ , = } . Z 2 is the theory whose axioms are: Peano Axioms, induction ( 0 ∈ X ∧ ∀ n ( n ∈ X → n + 1 ∈ X )) → ∀ n ( n ∈ X ) and the Comprehension Scheme ∃ X ∀ n ( n ∈ X ↔ ϕ ( n )) , with X not occurring free in ϕ . As we mentioned, in Reverse Mathematics we work with subsystems of Z 2 . Five of them turn out to be particularly important, and are called the Big Five. During this talk, we will mostly be concerned with the first three of them. April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 4 / 18

Reverse Mathematics Main subsystems RCA 0 is the L 2 -theory consisting of Peano axioms, the Σ 0 1 Induction Scheme ( ϕ ( 0 ) ∧ ∀ n ( ϕ ( n ) → ϕ ( n + 1 ))) → ∀ nϕ ( n ) , where ϕ is Σ 0 1 , and ∆ 0 1 Comprehension Scheme ∀ n ( ϕ ( n ) ↔ ψ ( n )) → ∃ X ∀ n ( n ∈ X ↔ ϕ ( n )) , where ϕ is Σ 0 1 , ψ is Π 0 1 and X does not occur free in ϕ . RCA 0 roughly corresponds to computable mathematics, and is used as a base system over which the equivalences between theorems and set existence axioms are proved. April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 5 / 18

Reverse Mathematics Main subsystems RCA 0 is the L 2 -theory consisting of Peano axioms, the Σ 0 1 Induction Scheme ( ϕ ( 0 ) ∧ ∀ n ( ϕ ( n ) → ϕ ( n + 1 ))) → ∀ nϕ ( n ) , where ϕ is Σ 0 1 , and ∆ 0 1 Comprehension Scheme ∀ n ( ϕ ( n ) ↔ ψ ( n )) → ∃ X ∀ n ( n ∈ X ↔ ϕ ( n )) , where ϕ is Σ 0 1 , ψ is Π 0 1 and X does not occur free in ϕ . RCA 0 roughly corresponds to computable mathematics, and is used as a base system over which the equivalences between theorems and set existence axioms are proved. April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 5 / 18

Reverse Mathematics Main subsystems (cont.) The other subsystems, ordered by incresing strength, that we will use in this talk are: WKL 0 : Weak K¨ onig’s Lemma, assert that every infinite binary tree has a path. Equivalent (over RCA 0 ) to Dilworth’s Theorem “every poset of width n can be partitioned into n chains” (see Hirst, 1987). ACA 0 : Arithmetical Comprehension Axiom, asserts the existence of every set that can be arithmetically defined. The other two main subsystems, which we will not use today, are: ATR 0 : Arithmetical Transfinite Recursion. Π 1 1 − CA 0 : Π 1 1 Comprehension Axiom. April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 6 / 18

Reverse Mathematics Main subsystems (cont.) The other subsystems, ordered by incresing strength, that we will use in this talk are: WKL 0 : Weak K¨ onig’s Lemma, assert that every infinite binary tree has a path. Equivalent (over RCA 0 ) to Dilworth’s Theorem “every poset of width n can be partitioned into n chains” (see Hirst, 1987). ACA 0 : Arithmetical Comprehension Axiom, asserts the existence of every set that can be arithmetically defined. The other two main subsystems, which we will not use today, are: ATR 0 : Arithmetical Transfinite Recursion. Π 1 1 − CA 0 : Π 1 1 Comprehension Axiom. April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 6 / 18

Reverse Mathematics Main subsystems (cont.) The other subsystems, ordered by incresing strength, that we will use in this talk are: WKL 0 : Weak K¨ onig’s Lemma, assert that every infinite binary tree has a path. Equivalent (over RCA 0 ) to Dilworth’s Theorem “every poset of width n can be partitioned into n chains” (see Hirst, 1987). ACA 0 : Arithmetical Comprehension Axiom, asserts the existence of every set that can be arithmetically defined. The other two main subsystems, which we will not use today, are: ATR 0 : Arithmetical Transfinite Recursion. Π 1 1 − CA 0 : Π 1 1 Comprehension Axiom. April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 6 / 18

Reverse Mathematics Main subsystems (cont.) The other subsystems, ordered by incresing strength, that we will use in this talk are: WKL 0 : Weak K¨ onig’s Lemma, assert that every infinite binary tree has a path. Equivalent (over RCA 0 ) to Dilworth’s Theorem “every poset of width n can be partitioned into n chains” (see Hirst, 1987). ACA 0 : Arithmetical Comprehension Axiom, asserts the existence of every set that can be arithmetically defined. The other two main subsystems, which we will not use today, are: ATR 0 : Arithmetical Transfinite Recursion. Π 1 1 − CA 0 : Π 1 1 Comprehension Axiom. April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 6 / 18

The principle RS - po Reverse Mathematics 1 The principle RS - po 2 Background on RS - po RS - po in Reverse Mathematics April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 7 / 18

The principle RS - po Background on RS - po Definitions and statement Recall that, given a poset ( P, < P ) : a chain C ⊂ P is a linearly ordered subset of P . an antichain A ⊂ P is a set such that ∀ a,b ∈ A ( a ≠ b → a ∣ P b ) , i.e. ∀ a,b ∈ A ( a ≠ b → a / ≤ P b ∧ b / ≤ P a ) the width of a poset P is the sup of the cardinalities of the antichains of P . Theorem (Rival-Sands) ( RS - po ) Let P be an infinite partial order of finite width. Then there exists an infinite chain C ⊂ P such that for every p ∈ P , p is comparable with 0 or infinitely many elements of C . April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 8 / 18

The principle RS - po Background on RS - po Rival-Sands for graphs One might wonder where such a statement comes from. The principle RS - po was introduced as a refinement of a result about graphs: Theorem (Rival and Sands) ( RS - g ) Let G be an infinite graph, then there exists an infinite subgraph H ⊂ G such that every vertex g ∈ G is adjacent to 0 , 1 or infinitely many vertices of H . Moreover, every h ∈ H is adjacent to 0 or infinitely many other elements of H . (one can be more precise about the relationship between the cardinality of G and that of H ) This result is interesting because it is, in some sense, complementary to Ramsey’s Theorem. April 25 th 2019 Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles 9 / 18