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

REVERSE MATHEMATICS OF SOME PRINCIPLES

RELATED TO PARTIAL ORDERS

Giovanni Sold` a, University of Leeds Joint work with Marta Fiori Carones, Alberto Marcone and Paul Shafer April 25th 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, Z2 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).

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 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, Z2 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).

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 2 / 18

Reverse Mathematics

1

Reverse Mathematics

2

The principle RS-po Background on RS-po RS-po in Reverse Mathematics

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 3 / 18

Reverse Mathematics

L2 and Z2

We will use the two-sorted (numbers and sets of numbers) language L2 = {0,1,+,⋅,<,∈,=}. Z2 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

• f Z2. 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.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 4 / 18

Reverse Mathematics

L2 and Z2

We will use the two-sorted (numbers and sets of numbers) language L2 = {0,1,+,⋅,<,∈,=}. Z2 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

• f Z2. 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.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 4 / 18

Reverse Mathematics

Main subsystems

RCA0 is the L2-theory consisting of Peano axioms, the Σ0

1 Induction

Scheme (ϕ(0) ∧ ∀n(ϕ(n) → ϕ(n + 1))) → ∀nϕ(n), where ϕ is Σ0

1, and ∆0 1 Comprehension Scheme

∀n(ϕ(n) ↔ ψ(n)) → ∃X∀n(n ∈ X ↔ ϕ(n)), where ϕ is Σ0

1, ψ is Π0 1 and X does not occur free in ϕ.

RCA0 roughly corresponds to computable mathematics, and is used as a base system over which the equivalences between theorems and set existence axioms are proved.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 5 / 18

Reverse Mathematics

Main subsystems

RCA0 is the L2-theory consisting of Peano axioms, the Σ0

1 Induction

Scheme (ϕ(0) ∧ ∀n(ϕ(n) → ϕ(n + 1))) → ∀nϕ(n), where ϕ is Σ0

1, and ∆0 1 Comprehension Scheme

∀n(ϕ(n) ↔ ψ(n)) → ∃X∀n(n ∈ X ↔ ϕ(n)), where ϕ is Σ0

1, ψ is Π0 1 and X does not occur free in ϕ.

RCA0 roughly corresponds to computable mathematics, and is used as a base system over which the equivalences between theorems and set existence axioms are proved.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 5 / 18

Reverse Mathematics

Main subsystems (cont.)

The other subsystems, ordered by incresing strength, that we will use in this talk are: WKL0: Weak K¨

• nig’s Lemma, assert that every infinite binary tree

has a path. Equivalent (over RCA0) to Dilworth’s Theorem “every poset of width n can be partitioned into n chains” (see Hirst, 1987). ACA0: Arithmetical Comprehension Axiom, asserts the existence

• f every set that can be arithmetically defined.

The other two main subsystems, which we will not use today, are: ATR0: Arithmetical Transfinite Recursion. Π1

1 − CA0: Π1 1 Comprehension Axiom.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 6 / 18

Reverse Mathematics

Main subsystems (cont.)

The other subsystems, ordered by incresing strength, that we will use in this talk are: WKL0: Weak K¨

• nig’s Lemma, assert that every infinite binary tree

has a path. Equivalent (over RCA0) to Dilworth’s Theorem “every poset of width n can be partitioned into n chains” (see Hirst, 1987). ACA0: Arithmetical Comprehension Axiom, asserts the existence

• f every set that can be arithmetically defined.

The other two main subsystems, which we will not use today, are: ATR0: Arithmetical Transfinite Recursion. Π1

1 − CA0: Π1 1 Comprehension Axiom.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 6 / 18

Reverse Mathematics

Main subsystems (cont.)

The other subsystems, ordered by incresing strength, that we will use in this talk are: WKL0: Weak K¨

• nig’s Lemma, assert that every infinite binary tree

has a path. Equivalent (over RCA0) to Dilworth’s Theorem “every poset of width n can be partitioned into n chains” (see Hirst, 1987). ACA0: Arithmetical Comprehension Axiom, asserts the existence

• f every set that can be arithmetically defined.

The other two main subsystems, which we will not use today, are: ATR0: Arithmetical Transfinite Recursion. Π1

1 − CA0: Π1 1 Comprehension Axiom.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 6 / 18

Reverse Mathematics

Main subsystems (cont.)

The other subsystems, ordered by incresing strength, that we will use in this talk are: WKL0: Weak K¨

• nig’s Lemma, assert that every infinite binary tree

has a path. Equivalent (over RCA0) to Dilworth’s Theorem “every poset of width n can be partitioned into n chains” (see Hirst, 1987). ACA0: Arithmetical Comprehension Axiom, asserts the existence

• f every set that can be arithmetically defined.

The other two main subsystems, which we will not use today, are: ATR0: Arithmetical Transfinite Recursion. Π1

1 − CA0: Π1 1 Comprehension Axiom.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 6 / 18

The principle RS-po

1

Reverse Mathematics

2

The principle RS-po Background on RS-po RS-po in Reverse Mathematics

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 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.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 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.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 9 / 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.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 9 / 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.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 9 / 18

The principle RS-po Background on RS-po

From graphs to posets

As Rival and Sands pointed out, the result above takes a much nicer form under the assumption that G is actually the comparability graph

• f a poset P of finite width.

With this setting in mind, we could rephrase RS-po as follows: Theorem If GP is the comparability graph of an infinite poset P of finite width, then there exists a complete subgraph H ⊂ GP such that every p ∈ P is adjacent to 0 or infinitely many elements of H. The theorem above is not, to the best of our knowledge, a trivial corollary of RS-g. (In this case, if one wants to be precise about the cardinalities of the infinite sets above, one must be more careful in the study of the relationship between the cardinality of P, its width and the cardinality

• f H)

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 10 / 18

The principle RS-po Background on RS-po

From graphs to posets

As Rival and Sands pointed out, the result above takes a much nicer form under the assumption that G is actually the comparability graph

• f a poset P of finite width.

With this setting in mind, we could rephrase RS-po as follows: Theorem If GP is the comparability graph of an infinite poset P of finite width, then there exists a complete subgraph H ⊂ GP such that every p ∈ P is adjacent to 0 or infinitely many elements of H. The theorem above is not, to the best of our knowledge, a trivial corollary of RS-g. (In this case, if one wants to be precise about the cardinalities of the infinite sets above, one must be more careful in the study of the relationship between the cardinality of P, its width and the cardinality

• f H)

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 10 / 18

The principle RS-po Background on RS-po

From graphs to posets

As Rival and Sands pointed out, the result above takes a much nicer form under the assumption that G is actually the comparability graph

• f a poset P of finite width.

With this setting in mind, we could rephrase RS-po as follows: Theorem If GP is the comparability graph of an infinite poset P of finite width, then there exists a complete subgraph H ⊂ GP such that every p ∈ P is adjacent to 0 or infinitely many elements of H. The theorem above is not, to the best of our knowledge, a trivial corollary of RS-g. (In this case, if one wants to be precise about the cardinalities of the infinite sets above, one must be more careful in the study of the relationship between the cardinality of P, its width and the cardinality

• f H)

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 10 / 18

The principle RS-po Background on RS-po

From graphs to posets

As Rival and Sands pointed out, the result above takes a much nicer form under the assumption that G is actually the comparability graph

• f a poset P of finite width.

With this setting in mind, we could rephrase RS-po as follows: Theorem If GP is the comparability graph of an infinite poset P of finite width, then there exists a complete subgraph H ⊂ GP such that every p ∈ P is adjacent to 0 or infinitely many elements of H. The theorem above is not, to the best of our knowledge, a trivial corollary of RS-g. (In this case, if one wants to be precise about the cardinalities of the infinite sets above, one must be more careful in the study of the relationship between the cardinality of P, its width and the cardinality

• f H)

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 10 / 18

The principle RS-po Background on RS-po

Remarks on the proof of RS-po in ZFC

The original proof of the theorem given by Rival and Sands actually gives a stronger result: Theorem (sRS-po) If P is an infinite poset of finite width, then there is a chain C of

• rder type ω or ω∗ such that every element p ∈ P is comparable with 0 or

infinitely many (and hence cofinitely many) elements of C. A direct translation of the original proof requires Π1

1 − CA0 to be

carried out (although by a standard result of Reverse Mathematics it cannot be that sRS-po and Π1

1 − CA0 are equivalent over RCA0).

We now present a proof of RS-po in ACA0.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 11 / 18

The principle RS-po Background on RS-po

Remarks on the proof of RS-po in ZFC

The original proof of the theorem given by Rival and Sands actually gives a stronger result: Theorem (sRS-po) If P is an infinite poset of finite width, then there is a chain C of

• rder type ω or ω∗ such that every element p ∈ P is comparable with 0 or

infinitely many (and hence cofinitely many) elements of C. A direct translation of the original proof requires Π1

1 − CA0 to be

carried out (although by a standard result of Reverse Mathematics it cannot be that sRS-po and Π1

1 − CA0 are equivalent over RCA0).

We now present a proof of RS-po in ACA0.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 11 / 18

The principle RS-po RS-po in Reverse Mathematics

Sketch of the proof in ACA0

Suppose for a contradiction that there is no solution. First of all, notice that if P contains a copy of Z, then that is a solution. Hence, we can suppose that this is not the case. We use Dilworth’s Theorem to decompose P into n disjoint chains C0,...,Cn−1 Lemma (ACA0) Every chain Ci as above can be separated into its well-founded and reverse-well-founded parts, W0,...,Wn−1 and R0,...,Rn−1. Suppose that W0,...,Wn−1 are all infinite and without maximum. We can find recursively a (cofinal) sequence B0 of type ω in W0. Since we are assuming neither B0 nor any of its tails are solutions, there is a counterexample B1 to it (not in B0), and it has order type ω. Iterating this argument n times, we run out of chains, thus obtaining a contradiction.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 12 / 18

The principle RS-po RS-po in Reverse Mathematics

Sketch of the proof in ACA0

Suppose for a contradiction that there is no solution. First of all, notice that if P contains a copy of Z, then that is a solution. Hence, we can suppose that this is not the case. We use Dilworth’s Theorem to decompose P into n disjoint chains C0,...,Cn−1 Lemma (ACA0) Every chain Ci as above can be separated into its well-founded and reverse-well-founded parts, W0,...,Wn−1 and R0,...,Rn−1. Suppose that W0,...,Wn−1 are all infinite and without maximum. We can find recursively a (cofinal) sequence B0 of type ω in W0. Since we are assuming neither B0 nor any of its tails are solutions, there is a counterexample B1 to it (not in B0), and it has order type ω. Iterating this argument n times, we run out of chains, thus obtaining a contradiction.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 12 / 18

The principle RS-po RS-po in Reverse Mathematics

Sketch of the proof in ACA0

Suppose for a contradiction that there is no solution. First of all, notice that if P contains a copy of Z, then that is a solution. Hence, we can suppose that this is not the case. We use Dilworth’s Theorem to decompose P into n disjoint chains C0,...,Cn−1 Lemma (ACA0) Every chain Ci as above can be separated into its well-founded and reverse-well-founded parts, W0,...,Wn−1 and R0,...,Rn−1. Suppose that W0,...,Wn−1 are all infinite and without maximum. We can find recursively a (cofinal) sequence B0 of type ω in W0. Since we are assuming neither B0 nor any of its tails are solutions, there is a counterexample B1 to it (not in B0), and it has order type ω. Iterating this argument n times, we run out of chains, thus obtaining a contradiction.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 12 / 18

The principle RS-po RS-po in Reverse Mathematics

Sketch of the proof in ACA0

Suppose for a contradiction that there is no solution. First of all, notice that if P contains a copy of Z, then that is a solution. Hence, we can suppose that this is not the case. We use Dilworth’s Theorem to decompose P into n disjoint chains C0,...,Cn−1 Lemma (ACA0) Every chain Ci as above can be separated into its well-founded and reverse-well-founded parts, W0,...,Wn−1 and R0,...,Rn−1. Suppose that W0,...,Wn−1 are all infinite and without maximum. We can find recursively a (cofinal) sequence B0 of type ω in W0. Since we are assuming neither B0 nor any of its tails are solutions, there is a counterexample B1 to it (not in B0), and it has order type ω. Iterating this argument n times, we run out of chains, thus obtaining a contradiction.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 12 / 18

The principle RS-po RS-po in Reverse Mathematics

Sketch of the proof in ACA0

Suppose for a contradiction that there is no solution. First of all, notice that if P contains a copy of Z, then that is a solution. Hence, we can suppose that this is not the case. We use Dilworth’s Theorem to decompose P into n disjoint chains C0,...,Cn−1 Lemma (ACA0) Every chain Ci as above can be separated into its well-founded and reverse-well-founded parts, W0,...,Wn−1 and R0,...,Rn−1. Suppose that W0,...,Wn−1 are all infinite and without maximum. We can find recursively a (cofinal) sequence B0 of type ω in W0. Since we are assuming neither B0 nor any of its tails are solutions, there is a counterexample B1 to it (not in B0), and it has order type ω. Iterating this argument n times, we run out of chains, thus obtaining a contradiction.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 12 / 18

The principle RS-po RS-po in Reverse Mathematics

Further results

Definition We call RS-pon the principle “If P is a poset of width n then the conclusion of RS-po holds for P”. ADS is the principle asserting that every infinite linear order contains a chain of order type ω or a chain of order type ω∗. SADS is the principle asserting that every infinite linear order of

• rder type ω + ω∗ contains a chain of type ω or a chain of type ω∗

ADS and SADS were introduced by Hirschfeldt and Shore (Hirschfeldt and Shore, 2007), in the study of Ramsey’s Theorem for pairs. When the width is fixed and less or equal than 3, the proof can be improved: Theorem RCA0 + ADS ⊢ RS-po3

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 13 / 18

The principle RS-po RS-po in Reverse Mathematics

Further results

Definition We call RS-pon the principle “If P is a poset of width n then the conclusion of RS-po holds for P”. ADS is the principle asserting that every infinite linear order contains a chain of order type ω or a chain of order type ω∗. SADS is the principle asserting that every infinite linear order of

• rder type ω + ω∗ contains a chain of type ω or a chain of type ω∗

ADS and SADS were introduced by Hirschfeldt and Shore (Hirschfeldt and Shore, 2007), in the study of Ramsey’s Theorem for pairs. When the width is fixed and less or equal than 3, the proof can be improved: Theorem RCA0 + ADS ⊢ RS-po3

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 13 / 18

The principle RS-po RS-po in Reverse Mathematics

The proof for width 2

We give an idea of how the proof of RS-po2 from ADS goes. Let P be a poset of width 2, and again we suppose that P contains no solution to RS-po2. In RCA0, we can find a linear order (LP ,<LP ) such that P = LP and for every p,q ∈ P,p <P q → p <LP q. Then, using ADS, we can find a chain B, say of type ω in (LP ,<LP ). It can be shown in RCA0 that B can be refined to B′ ∶= {b′

0 <P b′ 1 <P ...}, an infinite

<P -chain. It is still of type ω. Since neither B′ nor any of its tails can be a solution by our assumption, we can find a local counterexample to it: there are infinitely many indices i such that for some point di di >P b′

i and di∣P b′ i+1. We

collect (a subset of) the di’s into a set D, which can be refined to an ω-chain.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 14 / 18

The principle RS-po RS-po in Reverse Mathematics

The proof for width 2

We give an idea of how the proof of RS-po2 from ADS goes. Let P be a poset of width 2, and again we suppose that P contains no solution to RS-po2. In RCA0, we can find a linear order (LP ,<LP ) such that P = LP and for every p,q ∈ P,p <P q → p <LP q. Then, using ADS, we can find a chain B, say of type ω in (LP ,<LP ). It can be shown in RCA0 that B can be refined to B′ ∶= {b′

0 <P b′ 1 <P ...}, an infinite

<P -chain. It is still of type ω. Since neither B′ nor any of its tails can be a solution by our assumption, we can find a local counterexample to it: there are infinitely many indices i such that for some point di di >P b′

i and di∣P b′ i+1. We

collect (a subset of) the di’s into a set D, which can be refined to an ω-chain.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 14 / 18

The principle RS-po RS-po in Reverse Mathematics

The proof for width 2

We give an idea of how the proof of RS-po2 from ADS goes. Let P be a poset of width 2, and again we suppose that P contains no solution to RS-po2. In RCA0, we can find a linear order (LP ,<LP ) such that P = LP and for every p,q ∈ P,p <P q → p <LP q. Then, using ADS, we can find a chain B, say of type ω in (LP ,<LP ). It can be shown in RCA0 that B can be refined to B′ ∶= {b′

0 <P b′ 1 <P ...}, an infinite

<P -chain. It is still of type ω. Since neither B′ nor any of its tails can be a solution by our assumption, we can find a local counterexample to it: there are infinitely many indices i such that for some point di di >P b′

i and di∣P b′ i+1. We

collect (a subset of) the di’s into a set D, which can be refined to an ω-chain.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 14 / 18

The principle RS-po RS-po in Reverse Mathematics

The proof for width 2 (cont.)

Again, neither D nor any of its tails can be a solution, so we can find an ω-chain E local counterexample to D. If E >P B′, then C ∶= B′ ∪ E is a solution: intuitively, this is the case because we have already found infinitely many maximal antichains (the ones given by D and B′). Otherwise, it can be shown that a subset

• f B′ ∪ D is a solution: the idea is that this set os obtained by

“zig-zagging” between B′ and D.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 15 / 18

The principle RS-po RS-po in Reverse Mathematics

The proof for width 2 (cont.)

Again, neither D nor any of its tails can be a solution, so we can find an ω-chain E local counterexample to D. If E >P B′, then C ∶= B′ ∪ E is a solution: intuitively, this is the case because we have already found infinitely many maximal antichains (the ones given by D and B′). Otherwise, it can be shown that a subset

• f B′ ∪ D is a solution: the idea is that this set os obtained by

“zig-zagging” between B′ and D.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 15 / 18

The principle RS-po RS-po in Reverse Mathematics

The proof for width 2 (cont.)

Again, neither D nor any of its tails can be a solution, so we can find an ω-chain E local counterexample to D. If E >P B′, then C ∶= B′ ∪ E is a solution: intuitively, this is the case because we have already found infinitely many maximal antichains (the ones given by D and B′). Otherwise, it can be shown that a subset

• f B′ ∪ D is a solution: the idea is that this set os obtained by

“zig-zagging” between B′ and D.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 15 / 18

The principle RS-po RS-po in Reverse Mathematics

Remarks on the results

The proof of RS-po3 builds on the ideas already present the proof of RS-po2, basically by listing the possible behaviors of the chains of counterexamples. The generalization to width 4 (and more) appears to be highly nontrivial, and is still work in progress. There is also another proof of RS-po2, using a different set of axioms: Lemma WKL0 + SRT2

2 ⊢ RS-po2

where SRT2

2 is Stable Ramsey’s Theorem for pairs.

By a deep result of Chong, Yang and Slaman (Chong, Yang, and Slaman, 2010), the lemma above implies that RS-po2 cannot imply ADS

• ver RCA0.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 16 / 18

The principle RS-po RS-po in Reverse Mathematics

Remarks on the results

The proof of RS-po3 builds on the ideas already present the proof of RS-po2, basically by listing the possible behaviors of the chains of counterexamples. The generalization to width 4 (and more) appears to be highly nontrivial, and is still work in progress. There is also another proof of RS-po2, using a different set of axioms: Lemma WKL0 + SRT2

2 ⊢ RS-po2

where SRT2

2 is Stable Ramsey’s Theorem for pairs.

By a deep result of Chong, Yang and Slaman (Chong, Yang, and Slaman, 2010), the lemma above implies that RS-po2 cannot imply ADS

• ver RCA0.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 16 / 18

The principle RS-po RS-po in Reverse Mathematics

Remarks on the results

The proof of RS-po3 builds on the ideas already present the proof of RS-po2, basically by listing the possible behaviors of the chains of counterexamples. The generalization to width 4 (and more) appears to be highly nontrivial, and is still work in progress. There is also another proof of RS-po2, using a different set of axioms: Lemma WKL0 + SRT2

2 ⊢ RS-po2

where SRT2

2 is Stable Ramsey’s Theorem for pairs.

By a deep result of Chong, Yang and Slaman (Chong, Yang, and Slaman, 2010), the lemma above implies that RS-po2 cannot imply ADS

• ver RCA0.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 16 / 18

The principle RS-po RS-po in Reverse Mathematics

Some reversals

Although we do not have any equivalence for RS-po yet, we were able to prove the following implications: Theorem RCA0 + RS-po3 ⊢ ADS RCA0 + RS-po2 ⊢ SADS In particular, we have the following corollary: Corollary RCA0 ⊢ ADS ↔ RS-po3. RCA0 + RS-po2 / ⊢ RS-po3 Notably, this appears to be the first case of a principle coming from usual mathematics being proven to be equivalent to ADS.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 17 / 18

The principle RS-po RS-po in Reverse Mathematics

Some reversals

Although we do not have any equivalence for RS-po yet, we were able to prove the following implications: Theorem RCA0 + RS-po3 ⊢ ADS RCA0 + RS-po2 ⊢ SADS In particular, we have the following corollary: Corollary RCA0 ⊢ ADS ↔ RS-po3. RCA0 + RS-po2 / ⊢ RS-po3 Notably, this appears to be the first case of a principle coming from usual mathematics being proven to be equivalent to ADS.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 17 / 18

References

References

Chong, C. T., Y. Yang, and T. Slaman (2010). “The metamathematics of stable Ramsey’s theorem for pairs”. In: URL: https://math.berkeley.edu/˜slaman/papers/SRT22.pdf. Hirschfeldt, Denis R. and Richard A. Shore (2007). “Combinatorial principles weaker than Ramsey’s Theorem for pairs”. In: The Journal

• f Symbolic Logic 72.01, 171–206. DOI: 10.2178/jsl/1174668391.

Hirst, Jeffry L. (1987). “Combinatorics in Subsystems of Second order Arithmetic”. PhD thesis. The Pennsylvania State University. Rival, Ivan and Bill Sands (1980). “On the Adjacency of Vertices to the Vertices of an Infinite Subgraph”. In: Journal of the London Mathematical Society s2-21.3, 393–400. DOI: 10.1112/jlms/s2-21.3.393. Simpson, Stephen G. (2010). Subsystems of second order arithmetic. Cambridge Univ. Press.

Giovanni Sold` a, Leeds Reverse Math of Rival-Sands Principles April 25th 2019 18 / 18