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 L 2 has two sorts of variables: one 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 Z 2 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 L 2 has the form M = ( | M | , S M , 0 M , 1 M , + M , · M , < M ) where | M | serves as the range of the number variables, S M is a set of subsets of | M | serving as the range of the set variables. An ω -model is an L 2 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 L 2 . 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 RCA 0 : algebraic axioms for the natural numbers, Σ 0 1 -induction, and comprehension for ∆ 0 ω ω 1 formulas ω ω 2 WKL 0 = RCA 0 + K¨ onig’s lemma for binary trees 3 ACA 0 : comprehension extended to arithmetical formulas ε 0 4 ATR 0 = ACA 0 + defin. by arithmetical transfinite recursion Γ 0 5 Π 1 1 -CA 0 : comprehension extended to Π 1 1 formulas Ψ Ω 1 (Ω ω ) RCA 0 is the base theory for reverse mathematics: it allows the development of “computable mathematics”. RCA 0 and WKL 0 are Π 0 2 -conservative over PRA. ACA 0 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 RT 2 2 does not imply ACA 0 . It was already known that WKL 0 does not prove RT 2 2 . In 2012 Liu showed that RT 2 2 does not imply WKL 0 . After Seetapun’s result, many statements provable in ACA 0 , unprovable in RCA 0 , and not equivalent to either ACA 0 or WKL 0 , 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) ACA RT22+WKL+RRT32 RT22 PT22 SRT22+COH RWKL1 EM+ADS CAC+WKL SRAM POS+WWKL FS(3) IPT22 P22 CAC ASRAM 2-WWKL RRT32 FS(2) SRT22 EM BSig2+COH+RRT22 SCAC+CCAC ADS COH+WKL BSig2+Pi01G WKL 2-POS DTCp BSig2+2-RAN 2-RAN TS(2) SFS(2) SIPT22 SPT22 D22 SEM+SADS BSig2+RRT22 EM+BSig2 StCOH SCAC SADS+CADS CCAC Pi01G ISig2+AMT ISig2 WWKL ASRT22 POS RWKL RWWKL1 STS(2) SEM 1-GEN BSig2+CADS BSig2+COH CRT22+BSig2 StCRT22 StCADS SADS COH 1-POS 1-RAN RCOLOR3 RRT22 DNR DNR[0'] RCOLOR2 OPT AMT BSig2 CRT22 NCF RWWKL Kurtz CADS FIP PHPM PART RT1 AST nD2IP RCA Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 10 / 41
� � � Reverse mathematics Three beasts in the zoo 2 Ramsey Theorem for Pairs: for every f : [ N ] 2 → { 0 , 1 } RT 2 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 � RCA 0 ACA 0 WKL 0 � CAC � ADS RT 2 2 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 { q n | n ∈ N } ⊆ Q there exist i < j such that q i ≤ Q q j . 2 Q is wqo(set) if for every { q n | n ∈ N } ⊆ Q there exists A ⊆ N infinite such that q i ≤ Q q j , 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. WKL 0 + 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 RCA 0 wqo(fbp) wqo(anti) � wqo wqo(set) 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 RCA 0 . Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 14 / 41
� � � � � � Definitions of wqo The implications provable in WKL 0 wqo wqo(set) wqo(anti) wqo(ext) There are ω -models showing that the missing implications are not provable in WKL 0 . Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 15 / 41
� � � � � Definitions of wqo The implications provable in RCA 0 + 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 RCA 0 , 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 RCA 0 , the following are equivalent: 1 Π 1 1 -CA 0 ; 2 the minimal bad sequence lemma. This entails that the usual proofs of Higman’s and Kruskal’s theorems can be carried out in Π 1 1 -CA 0 . Alberto Marcone (Universit` a di Udine) Wqo and bqo theory in reverse mathematics 19 / 41
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