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Wqo and bqo theory in reverse mathematics

Alberto Marcone

Dipartimento di Scienze Matematiche, Informatiche e Fisiche Universit` a di Udine Italy alberto.marcone@uniud.it http://www.dimi.uniud.it/marcone

Well Quasi-Orders in Computer Science January 17–22, 2016 Shloss Dagstuhl

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 1 / 41

Outline

1 Reverse mathematics 2 Definitions of wqo 3 Examples and closure properties of wqos 4 Maximal linear extensions and maximal chains in wqos 5 From wqos to Noetherian topologies 6 Bqos

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 2 / 41

Reverse mathematics

Reverse mathematics

1 Reverse mathematics 2 Definitions of wqo 3 Examples and closure properties of wqos 4 Maximal linear extensions and maximal chains in wqos 5 From wqos to Noetherian topologies 6 Bqos

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 3 / 41

Reverse mathematics

Reverse mathematics

The goal of reverse mathematics is to establish which set existence axioms are required

• to prove theorems of ordinary mathematics;
• to show that different definitions of the same ordinary mathematics

concept are equivalent. This happens mostly in the context of subsystems of second order arithmetic.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 4 / 41

Reverse mathematics

Second order arithmetic

The language of second order arithmetic L2 has two sorts of variables:

• ne for natural numbers, the other for sets of natural numbers.

There are symbols for basic algebraic operations, for equality between natural numbers, and for membership between a number and a set. We use classical logic. Full second order arithmetic Z2 is the theory with algebraic axioms for the natural numbers, full induction, and full comprehension: ∃X ∀n (n ∈ X ⇐ ⇒ ϕ(n)), with X not free in ϕ(n).

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 5 / 41

Reverse mathematics

Semantics of second order arithmetic

A model for L2 has the form M = (|M|, SM, 0M, 1M, +M, ·M, <M) where |M| serves as the range of the number variables, SM is a set of subsets of |M| serving as the range of the set variables. An ω-model is an L2 model M whose first order part is standard, i.e. of the form (ω, 0, 1, +, ·, <). Thus M can be identified with the collection of sets of natural numbers serving as the range of the set variables in L2. For example REC is the ω-model consisting of the computable (or recursive) sets.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 6 / 41

Reverse mathematics

Mathematics in second order arithmetic

The idea that in (subsystems of) second order arithmetic it is possible to state and prove many significant mathematical theorems goes back to Hermann Weyl, Hilbert and Bernays. The systematic search for the subsystems of second order arithmetic which are sufficient and necessary to prove these theorems was started by Harvey Friedman around 1970, and pursued by Steve Simpson and many others.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 7 / 41

Reverse mathematics

Subsystems of second order arithmetic The big five of reverse mathematics

1 RCA0: algebraic axioms for the natural numbers,

Σ0

1-induction, and comprehension for ∆0 1 formulas

ωω

2 WKL0 = RCA0+ K¨

• nig’s lemma for binary trees

ωω

3 ACA0: comprehension extended to arithmetical formulas

ε0

4 ATR0 = ACA0+ defin. by arithmetical transfinite recursion

Γ0

5 Π1 1-CA0: comprehension extended to Π1 1 formulas

ΨΩ1(Ωω) RCA0 is the base theory for reverse mathematics: it allows the development of “computable mathematics”. RCA0 and WKL0 are Π0

2-conservative over PRA.

ACA0 is conservative over PA.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 8 / 41

Reverse mathematics

The zoo

In 1995 Seetapun showed that Ramsey Theorem for pairs RT2

2 does not

imply ACA0. It was already known that WKL0 does not prove RT2

2.

In 2012 Liu showed that RT2

2 does not imply WKL0.

After Seetapun’s result, many statements provable in ACA0, unprovable in RCA0, and not equivalent to either ACA0 or WKL0, have been discovered. Computability theory constructions based on forcing are used to build ω-models of one statement but not of the other. The neat five-levels building of XXth century reverse mathematics is now much more complex, with lots of different beasts: the zoo of XXIst century reverse mathematics.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 9 / 41

Reverse mathematics

A picture of the zoo (by Ludovic Patey)

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 10 / 41

Reverse mathematics

Three beasts in the zoo

RT2

2 Ramsey Theorem for Pairs: for every f : [N]2 → {0, 1}

there exist i < 2 and H ⊆ N infinite such that f(n, m) = i for every n, m ∈ H CAC Chain-Antichain: every infinite partial order has either an infinite chain or an infinite antichain ADS Ascending Sequence-Descending Sequence: every infinite linear order has either an infinite ascending sequence or an infinite descending sequence ACA0

• WKL0

RCA0

RT2

2

CAC ADS

• Alberto Marcone (Universit`

a di Udine) Wqo and bqo theory in reverse mathematics 11 / 41

Definitions of wqo

Definitions of wqo

1 Reverse mathematics 2 Definitions of wqo 3 Examples and closure properties of wqos 4 Maximal linear extensions and maximal chains in wqos 5 From wqos to Noetherian topologies 6 Bqos

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 12 / 41

Definitions of wqo

Five different definitions of wqo

Let Q = (Q, ≤Q) be a qo.

1 Q is wqo if for every { qn | n ∈ N } ⊆ Q there exist i < j such that

qi ≤Q qj.

2 Q is wqo(set) if for every { qn | n ∈ N } ⊆ Q there exists A ⊆ N

infinite such that qi ≤Q qj, for every i < j with i, j ∈ A.

3 Q is wqo(anti) if it is well-founded and has no infinite antichain. 4 Q is wqo(ext) if each of its linear extensions is well ordered. 5 Q is wqo(fbp) if for every X ⊆ Q there exists a finite F ⊆ X such

that ∀x ∈ X ∃y ∈ F (y ≤Q x). In (Cholak-M-Solomon 2004) we started the study of the reverse mathematics of the implications between these characterizations. WKL0 + CAC suffices to prove that the five notions are equivalent.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 13 / 41

Definitions of wqo

The implications provable in RCA0

wqo(fbp)

• wqo(anti)

wqo(set)

wqo

• wqo(ext)

There are computable counterexamples to the missing arrows. These are exactly the implications that are

• true in the ω-model REC,
• provable in RCA0.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 14 / 41

Definitions of wqo

The implications provable in WKL0

wqo

• wqo(set)
• wqo(anti)

wqo(ext)

• There are ω-models showing that the missing implications are not provable

in WKL0.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 15 / 41

Definitions of wqo

The implications provable in RCA0 + CAC

wqo(set)

• wqo
• wqo(anti)
• wqo(ext)

It remains open whether the missing arrows hold.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 16 / 41

Definitions of wqo

A few reverse mathematics results

Theorem Within RCA0, the following are equivalent:

1 CAC; 2 wqo(anti) → wqo(set) (Cholak-M-Solomon 2004); 3 wqo(anti) → wqo(exp) (Frittaion 2014); 4 wqo(anti) → wqo (Frittaion 2014); 5 wqo → wqo(set) (Frittaion 2014).

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 17 / 41

Examples and closure properties of wqos

Examples and closure properties of wqos

1 Reverse mathematics 2 Definitions of wqo 3 Examples and closure properties of wqos 4 Maximal linear extensions and maximal chains in wqos 5 From wqos to Noetherian topologies 6 Bqos

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 18 / 41

Examples and closure properties of wqos

The minimal bad sequence lemma

Theorem (M-Simpson 1996) Within RCA0, the following are equivalent:

1 Π1 1-CA0; 2 the minimal bad sequence lemma.

This entails that the usual proofs of Higman’s and Kruskal’s theorems can be carried out in Π1

1-CA0.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 19 / 41

Examples and closure properties of wqos

Some examples of wqos

Fact RCA0 proves that finite qos and well orders are wqos. Theorem (Simpson 1988) Within RCA0, the following are equivalent:

1 embeddability on finite strings over a finite alphabet is a wqo; 2 ωωω is well-founded.

Theorem (Friedman 1985) Kruskal’s theorem asserting that embeddability on finite trees is a wqo implies that the ordinal Γ0 is well-founded. Thus ATR0 does not prove Kruskal’s theorem. The exact strength of Kruskal’s theorem was established in (Rathjen-Weiermann 1993).

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 20 / 41

Examples and closure properties of wqos

The minor graph theorem

Theorem (Friedman-Robertson-Seymour 1987) The Graph Minor Theorem asserting that the minor relation on finite graphs is a wqo implies that the ordinal ΨΩ1(Ωω) is well-founded. Thus Π1

1-CA0 does not prove the Graph Minor Theorem.

Notice that this result predates of about 15 years Robertson and Seymour’s last paper in the proof of the Graph Minor Theorem. Some upper bound on the strength of the Graph Minor Theorem has been recently obtained by Michael Rathjen and his student Martin Krombholz.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 21 / 41

Examples and closure properties of wqos

The complexity of statements

The wqo theorems we considered until now are Π1

1 statements,

i.e. of the form ∀X ϕ(X) where ϕ has no set quantifiers. Hence they do not assert the existence of any sets and are true in REC (in fact in any ω-model). They can only imply well-foundedness assertions. To obtain set existence axioms we need to look at Π1

2 statements,

i.e. of the form ∀X ∃Y ψ(X, Y ) where ψ has no set quantifiers. Model-theoretic considerations show that no true Π1

2 statement implies

Π1

1-CA0.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 22 / 41

Examples and closure properties of wqos

Fra¨ ıss´ e’s conjecture

Fra¨ ıss´ e’s Conjecture is the statement that embeddability on countable linear orders is a wqo. We keep it distinct from Laver’s Theorem, i.e. the stronger statement that embeddability on countable linear orders is a bqo. Notice that the embeddability relation is more complex (i.e. Σ1

1) than in

the previous examples. Thus the statement is Π1

2.

Fact Π1

2-CA0 proves Laver’s Theorem and hence Fra¨

ıss´ e’s Conjecture. Theorem (Shore 1993) Fra¨ ıss´ e’s Conjecture implies ATR0.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 23 / 41

Examples and closure properties of wqos

A restriction of Fra¨ ıss´ e’s Conjecture

To gain a better understanding of the strength of Fra¨ ıss´ e’s Conjecture, we restricted the class of countable linear orders. Theorem (M-Montalb´ an 2009) ACA+

0 + ϕ2(0) is well-founded proves Fra¨

ıss´ e’s Conjecture restricted to linear orders of finite Hausdorff rank. Fra¨ ıss´ e’s Conjecture restricted to linear orders of finite Hausdorff rank implies ACA′

0 + ϕ2(0) is well-founded.

Here ϕ2(0) is the least fixed point of the function α → εα. These theories are much weaker than ATR0.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 24 / 41

Examples and closure properties of wqos

Basic closure properties of wqos

Fact RCA0 proves that the sum and the disjoint sum of two wqos are wqos. Lemma (M 2005) Within RCA0, the following are equivalent:

1 the product of two wqos is a wqo; 2 the intersection of two wqos (on the same set) is a wqo.

Theorem (Cholak-M-Solomon 2004) CAC proves that the product of two wqos is a wqo, but WKL0 does not. Two weeks ago at the reverse mathematics workshop in Singapore we noticed that “the product of two wqos is a wqo” implies ADS.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 25 / 41

Examples and closure properties of wqos

Higman’s Theorem

Theorem (Simpson 1988) Within RCA0, the following are equivalent:

1 ACA0; 2 Higman’s Theorem: embeddability on finite sequences from a wqo is

a wqo. The proof of 1 = ⇒ 2 avoids the minimal bad sequence lemma by using the reification of a wqo by a well order.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 26 / 41

Examples and closure properties of wqos

A structure theorem for wqos

Definition A ⊆ Q is an ideal if it is downward closed and ∀p, q ∈ A ∃r ∈ A(p ≤Q r ∧ q ≤Q r). Theorem (Erd˝

• s-Tarski 1943, Bonnet 1975)

Q has no infinite antichain if and only if every downward closed subset of Q is a finite union of ideals. Theorem (Frittaion-M 2014) Within RCA0, the following are equivalent:

1 ACA0; 2 every wqo is a finite union of ideals; 3 every partial order with no infinite antichain is a finite union of ideals.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 27 / 41

Maximal linear extensions and maximal chains in wqos

Maximal linear extensions and maximal chains in wqos

1 Reverse mathematics 2 Definitions of wqo 3 Examples and closure properties of wqos 4 Maximal linear extensions and maximal chains in wqos 5 From wqos to Noetherian topologies 6 Bqos

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 28 / 41

Maximal linear extensions and maximal chains in wqos

The maximal linear extension theorem

If Q is a wqo we denote by Lin(Q) the set of all linear extensions of Q. Definition If Q is a wqo, its maximal order type is

• (Q) = sup{ α | ∃L ∈ Lin(Q) α = o. t.(L) }.

Theorem (de Jongh-Parikh, 1977) The sup in the definition of o(Q) is actually a max. Theorem (M-Shore 2011) Within RCA0, the following are equivalent:

1 ATR0; 2 if Q is a wqo there exists L ∈ Lin(Q) such that I L for all

I ∈ Lin(Q);

3 the same restricted to disjoint unions of two linear orders.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 29 / 41

Maximal linear extensions and maximal chains in wqos

Maximal order types in reverse mathematics

Maximal order types are a useful tool to calibrate the strength of statements about wqo.

• the maximal order type of the wqo of finite trees is used prove that

Kruskal’s theorem cannot be proved in ATR0 (Friedman 1985);

• the maximal order type of the wqo of finite graphs is used prove that

the Robertson-Seymour theorem cannot be proved in Π1

1-CA0

(Friedman-Robertson-Seymour 1987);

• the maximal order type of certain wqos is used to establish the

strength of Higman’s theorem and of the Hilbert basis theorem (Simpson 1988);

• the maximal order type of the wqo of scattered linear orders of finite

Hausdorff rank is used to establish the strength of the restriction of Fra¨ ıss´ e’s conjecture to those linear orders (M-Montalb´ an 2009).

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 30 / 41

Maximal linear extensions and maximal chains in wqos

The maximal chain theorem

Q is well-founded we denote by Ch(Q) the collection of all chains in Q. Definition If Q is well-founded, its height is h(Q) = sup{ α | ∃C ∈ Ch(Q) α = o. t.(C) }. Theorem (Wolk 1967) If Q is a wqo, the sup in the definition of h(Q) is actually a max. Theorem (M-Shore 2011) Within RCA0, the following are equivalent:

1 ATR0; 2 if Q is a wqo there exists C ∈ Ch(Q) such that C′ C for all

C′ ∈ Ch(Q);

3 the same restricted to disjoint unions of two linear orders.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 31 / 41

From wqos to Noetherian topologies

From wqos to Noetherian topologies

1 Reverse mathematics 2 Definitions of wqo 3 Examples and closure properties of wqos 4 Maximal linear extensions and maximal chains in wqos 5 From wqos to Noetherian topologies 6 Bqos

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 32 / 41

From wqos to Noetherian topologies

Quasi-orders on the powerset

Let Q = (Q, ≤Q) be a quasi-order. For A, B ∈ P(Q): A ≤♭ B ⇐ ⇒ ∀a ∈ A ∃b ∈ B a ≤Q b ⇐ ⇒ A ⊆ B ↓ A ≤♯ B ⇐ ⇒ ∀b ∈ B ∃a ∈ A a ≤Q b ⇐ ⇒ B ⊆ A ↑ Let P♭(Q) = (P(Q), ≤♭) and P♯(Q) = (P(Q), ≤♯). P♭

f (Q) and P♯ f (Q) are the restrictions to finite subsets of Q.

If Q is wqo then P♭

f (Q) is wqo (Erd˝

• s-Rado 1952),

but P♭(Q), P♯(Q) and P♯

f (Q) are in general not wqo.

Theorem (Frittaion-Hendtlass-M-Shafer-Van der Meeren 2016) Within RCA0, the following are equivalent:

1 ACA0; 2 if Q is wqo, then P♭ f (Q) is wqo.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 33 / 41

From wqos to Noetherian topologies

Noetherian spaces

A topological space X is Noetherian if every open subset of X is compact. Some equivalent characterizations of Noetherian spaces:

• every subset of X is compact;
• every increasing sequence of open subsets of X stabilizes;
• every decreasing sequence of closed subsets of X stabilizes.

Noetherian spaces are important in algebraic geometry: the set of prime ideals (aka the spectrum) of a Noetherian ring with the Zariski topology is a Noetherian space. If a T2 space is Noetherian then it is finite.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 34 / 41

From wqos to Noetherian topologies

From quasi-orders to topological spaces

Let Q = (Q, ≤Q) be a quasi-order. The Alexandroff topology A(Q) is the topology on Q with the downward closed subsets of Q as closed sets. The upper topology U(Q) is the topology on Q with the downward closures of finite subsets of Q as a basis for the closed sets. Why these two topologies? Given a topological space, define a quasi-order on the points by x y ⇐ ⇒ every open set that contains x also contains y. A(Q) is the finest topology on Q such that is ≤Q. U(Q) is the coarsest such topology. If Q is not an antichain A(Q) and U(Q) are not T1.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 35 / 41

From wqos to Noetherian topologies

Which features of the quasi-order Q are reflected in A(Q) and U(Q)?

Fact Q is wqo if and only if A(Q) is Noetherian. If Q is wqo then U(Q) is Noetherian. Recall that by Erd˝

• s and Rado if Q is wqo, then P♭

f (Q) is a wqo.

Thus if Q is wqo, then U(P♭

f (Q)) is Noetherian.

However U(Q) might be Noetherian even when Q is not wqo. Theorem (Goubault-Larrecq, 2007) If Q is wqo then U(P♭(Q)), U(P♯

f (Q)) and U(P♯(Q)) are Noetherian.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 36 / 41

From wqos to Noetherian topologies

The reverse mathematics of Goubault-Larrecq’s theorems

Theorem (Frittaion-Hendtlass-M-Shafer-Van der Meeren 2016) Within RCA0, the following are equivalent:

1 ACA0; 2 if Q is wqo then A(P♭ f (Q)) is Noetherian; 3 if Q is wqo then U(P♭ f (Q)) is Noetherian; 4 if Q is wqo then U(P♯ f (Q)) is Noetherian; 5 if Q is wqo then U(P♭(Q)) is Noetherian; 6 if Q is wqo then U(P♯(Q)) is Noetherian.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 37 / 41

Bqos

Bqos

1 Reverse mathematics 2 Definitions of wqo 3 Examples and closure properties of wqos 4 Maximal linear extensions and maximal chains in wqos 5 From wqos to Noetherian topologies 6 Bqos

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 38 / 41

Bqos

Some examples of bqos

Fact RCA0 proves that well orders and any partial order with 2 elements are bqos. ATR0 proves that any finite qo is bqo. Lemma (M 2005) For any n ≥ 3, RCA0 proves that if the 3 elements antichain is a bqo then any partial order with n elements is a bqo. Question What is the strength of “the 3 elements antichain is a bqo”? Notice that “the 3 elements antichain is a bqo” is a consequence of “every interval order which is wqo is bqo”.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 39 / 41

Bqos

Laver’s Theorem

The main question is the strength of Laver’s Theorem, stating that embeddability on countable linear orders is a bqo. We know that Π1

2-CA0 proves it, that it implies ATR0,

and that it does not imply Π1

1-CA0.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 40 / 41

Bqos

Closure properties of bqos

Fact RCA0 proves that the sum of two bqos is bqo. Lemma (M 2005) ATR0 proves that the disjoint sum and the product of two bqos are bqos. Each of these two statements implies ACA0. Nash-Williams Theorem NWT: embeddability on countable sequences from a bqo is a bqo (Nash-Williams 1968). Theorem (M 1996)

• Π1

1-CA0 proves NWT;

• NWT implies ATR0;
• NWT does not imply Π1

1-CA0.

Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 41 / 41