Reverse Mathematics of some principles related to Partial ... · theory Aand a theorem T(formalized...

REVERSE MATHEMATICS OF SOME PRINCIPLESRELATED TO PARTIAL ORDERS

Giovanni Solda, University of Leeds

Joint work with Marta Fiori Carones, Alberto Marconeand Paul Shafer

April 25th 2019

Introduction & plan of the talk

Reverse Mathematics is an ongoing project whose goal is to measurethe “logical strength” of the theorems of ordinary mathematics: theclassification proceeds by analyzing the set existence axioms requiredto prove the theorems. As we will see during the talk, the typical“reverse mathematical” process goes as follows: given a weak basetheory 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 aboutthe combinatorics of infinite posets due to Rival and Sands (Rivaland Sands, 1980).

Giovanni Solda, 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 measurethe “logical strength” of the theorems of ordinary mathematics: theclassification proceeds by analyzing the set existence axioms requiredto prove the theorems. As we will see during the talk, the typical“reverse mathematical” process goes as follows: given a weak basetheory 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 aboutthe combinatorics of infinite posets due to Rival and Sands (Rivaland Sands, 1980).

Reverse Mathematics

1 Reverse Mathematics

2 The principle RS-poBackground on RS-poRS-po in Reverse Mathematics

Reverse Mathematics

L2 and Z2

We will use the two-sorted (numbers and sets of numbers) languageL2 = {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 subsystemsof Z2. Five of them turn out to be particularly important, and are calledthe Big Five. During this talk, we will mostly be concerned with thefirst three of them.

Reverse Mathematics

L2 and Z2

We will use the two-sorted (numbers and sets of numbers) languageL2 = {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 subsystemsof Z2. Five of them turn out to be particularly important, and are calledthe Big Five. During this talk, we will mostly be concerned with thefirst three of them.

Reverse Mathematics

Main subsystems

RCA0 is the L2-theory consisting of Peano axioms, the Σ01 Induction

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

where ϕ is Σ01, and ∆0

1 Comprehension Scheme

∀n(ϕ(n)↔ ψ(n))→ ∃X∀n(n ∈X ↔ ϕ(n)),

where ϕ is Σ01, ψ is Π0

1 and X does not occur free in ϕ.RCA0 roughly corresponds to computable mathematics, and is used asa base system over which the equivalences between theorems and setexistence axioms are proved.

Reverse Mathematics

Main subsystems

RCA0 is the L2-theory consisting of Peano axioms, the Σ01 Induction

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

where ϕ is Σ01, and ∆0

1 Comprehension Scheme

∀n(ϕ(n)↔ ψ(n))→ ∃X∀n(n ∈X ↔ ϕ(n)),

where ϕ is Σ01, ψ is Π0

1 and X does not occur free in ϕ.RCA0 roughly corresponds to computable mathematics, and is used asa base system over which the equivalences between theorems and setexistence axioms are proved.

Reverse Mathematics

Main subsystems (cont.)

The other subsystems, ordered by incresing strength, that we will usein this talk are:

WKL0: Weak Konig’s Lemma, assert that every infinite binary treehas a path. Equivalent (over RCA0) to Dilworth’s Theorem “everyposet of width n can be partitioned into n chains” (see Hirst, 1987).ACA0: Arithmetical Comprehension Axiom, asserts the existenceof 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: Π11 Comprehension Axiom.

Reverse Mathematics

The principle RS-po

1 Reverse Mathematics

2 The principle RS-poBackground on RS-poRS-po in Reverse Mathematics

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 theantichains of P .

Theorem (Rival-Sands)

(RS-po) Let P be an infinite partial order of finite width. Then there exists aninfinite chain C ⊂ P such that for every p ∈ P , p is comparable with 0 orinfinitely many elements of C.

Rival-Sands for graphs

One might wonder where such a statement comes from. The principleRS-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 subgraphH ⊂ G such that every vertex g ∈ G is adjacent to 0, 1 or infinitely manyvertices 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 thecardinality of G and that of H)This result is interesting because it is, in some sense, complementary toRamsey’s Theorem.

From graphs to posets

As Rival and Sands pointed out, the result above takes a much nicerform under the assumption that G is actually the comparability graphof a poset P of finite width.With this setting in mind, we could rephrase RS-po as follows:

TheoremIf GP is the comparability graph of an infinite poset P of finite width, thenthere exists a complete subgraph H ⊂ GP such that every p ∈ P is adjacent to0 or infinitely many elements of H .

The theorem above is not, to the best of our knowledge, a trivialcorollary of RS-g.(In this case, if one wants to be precise about the cardinalities of theinfinite sets above, one must be more careful in the study of therelationship between the cardinality of P , its width and the cardinalityof H)

Remarks on the proof of RS-po in ZFC

The original proof of the theorem given by Rival and Sands actuallygives a stronger result:

Theorem(sRS-po) If P is an infinite poset of finite width, then there is a chain C oforder type ω or ω∗ such that every element p ∈ P is comparable with 0 orinfinitely many (and hence cofinitely many) elements of C.

A direct translation of the original proof requires Π11 − CA0 to be

carried out (although by a standard result of Reverse Mathematics itcannot be that sRS-po and Π1

1 − CA0 are equivalent over RCA0).We now present a proof of RS-po in ACA0.

Remarks on the proof of RS-po in ZFC

The original proof of the theorem given by Rival and Sands actuallygives a stronger result:

Theorem(sRS-po) If P is an infinite poset of finite width, then there is a chain C oforder type ω or ω∗ such that every element p ∈ P is comparable with 0 orinfinitely many (and hence cofinitely many) elements of C.

A direct translation of the original proof requires Π11 − CA0 to be

carried out (although by a standard result of Reverse Mathematics itcannot be that sRS-po and Π1

1 − CA0 are equivalent over RCA0).We now present a proof of RS-po in ACA0.

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’sTheorem to decompose P into n disjoint chains C0, . . . ,Cn−1

Lemma(ACA0) Every chain Ci as above can be separated into its well-founded andreverse-well-founded parts, W0, . . . ,Wn−1 and R0, . . . ,Rn−1.

Suppose that W0, . . . ,Wn−1 are all infinite and without maximum. Wecan find recursively a (cofinal) sequence B0 of type ω in W0. Since weare assuming neither B0 nor any of its tails are solutions, there is acounterexample B1 to it (not in B0), and it has order type ω. Iterating thisargument n times, we run out of chains, thus obtaining a contradiction.

Further results

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

ADS and SADS were introduced by Hirschfeldt and Shore (Hirschfeldtand 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 beimproved:

TheoremRCA0 +ADS ⊢ RS-po3

Further results

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

ADS and SADS were introduced by Hirschfeldt and Shore (Hirschfeldtand 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 beimproved:

TheoremRCA0 +ADS ⊢ RS-po3

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 nosolution to RS-po2. In RCA0, we can find a linear order (LP ,<LP

) suchthat 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 inRCA0 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 ourassumption, we can find a local counterexample to it: there are infinitelymany indices i such that for some point di di >P b′i and di∣P b′i+1. Wecollect (a subset of) the di’s into a set D, which can be refined to anω-chain.

∶= {b′0 <P b′

The proof for width 2 (cont.)

Again, neither D nor any of its tails can be a solution, so we can findan 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 subsetof B′

∪D is a solution: the idea is that this set os obtained by“zig-zagging” between B′ and D.

Remarks on the results

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

WKL0 + SRT22 ⊢ RS-po2

where SRT22 is Stable Ramsey’s Theorem for pairs.

By a deep result of Chong, Yang and Slaman (Chong, Yang, andSlaman, 2010), the lemma above implies that RS-po2 cannot imply ADSover RCA0.

Some reversals

Although we do not have any equivalence for RS-po yet, we were ableto prove the following implications:

TheoremRCA0 + 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 fromusual mathematics being proven to be equivalent to ADS.

Some reversals

Although we do not have any equivalence for RS-po yet, we were ableto prove the following implications:

TheoremRCA0 + 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 fromusual mathematics being proven to be equivalent to ADS.

References

Chong, C. T., Y. Yang, and T. Slaman (2010). “The metamathematics ofstable Ramsey’s theorem for pairs”. In: URL:https://math.berkeley.edu/˜slaman/papers/SRT22.pdf.

Hirschfeldt, Denis R. and Richard A. Shore (2007). “Combinatorialprinciples weaker than Ramsey’s Theorem for pairs”. In: The Journalof Symbolic Logic 72.01, 171–206. DOI: 10.2178/jsl/1174668391.

Hirst, Jeffry L. (1987). “Combinatorics in Subsystems of Second orderArithmetic”. PhD thesis. The Pennsylvania State University.

Rival, Ivan and Bill Sands (1980). “On the Adjacency of Vertices to theVertices of an Infinite Subgraph”. In: Journal of the LondonMathematical 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.

Reverse Mathematics of some principles related to Partial ... · theory Aand a theorem T(formalized...

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