Newdirectionsinreversemathematics

DamirD.DzhafarovUniversityofCalifornia, Berkeley

February20, 2013

A foundationalmotivation.

Reversemathematics ismotivatedbyafoundationalquestion:

Question. Whichaxiomsdowe really needtoproveagiventheorem?

Thequestionleadstotheideaofthe strength ofatheorem. Whichtheoremsdoesitimply? Whichimplyit? Whichisitequivalentto?

Example. Over ZF, theaxiomofchoiceisequivalenttoZorn’slemma.

Settheoryismuchtoopowerfultoaccuratelycalibratestrength.

Wewouldlikeresultsoftheform

Overthetheory T, theorem P implies/isequivalenttotheorem Q,

where T isrelatively weak, butalso expressive enoughtoaccommodateadecentamountofmathematics.

Subsystemsofsecond-orderarithmetic.

Second-orderarithmetic, Z2, isatwo-sortedtheorywithvariablesfornumbers and setsofnumbers, andtheusual symbolsofarithmetic.

Theaxiomsof Z2 arethoseofPeanoarithmetic, andthe comprehensionscheme: if φ isaformula(withsetparameters)then {x : φ(x)} exists.

Example. If G0,G1, . . . and G aregroups, then {i ∈ N : Gi ∼= G} exists.

Werestrictwhichformulas φ weallowtoobtainvarious subsystems:

RCA0 ∆01 (computable)formulas.

WKL0 Formulasdefiningpathsthroughinfinitebinarytrees.ACA0 Arithmeticalformulas.

Twoothersubsystems, ATR0 and Π11-CA0, completethe “bigfive”.

Subsystemsofsecond-orderarithmetic.

Z2

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Connectionswithcomputabilitytheory.

A set A iscomputablefrom B, written A ⩽T B, if A = Φ(B) forsomeeffectiveprocedure Φ; i.e., knowing B, wecancomputablyfigureout A.

The haltingsetof A is A ′ = {i ∈ N : the ithprogramwithoracle A halts}.

A modelof Z2 consistsof(apossiblynonstandardversionof) N andasubsetof P(N) closedundercertaincomputability-theoreticoperations.

RCA0 Modelsclosedunder ⩽T and ⊕ (disjointunion).ACA0 Modelsclosedunderthejumpoperator, A 7→ A ′.

Thestrengthofatheoremcorrespondstohoweffectively solutions canbefoundtoagivencomputable instance ofthattheorem.

Example. DoeseverycomputablecommutativeunitringhaveaExample. computablemaximalideal? Doallidealscompute ∅ ′?

Reversemathematics.

Most(countable)classicalmathematicscanbedevelopedwithin Z2.

Theorem. Thefollowingareprovablein RCA0.

1(Simpson). Bairecategorytheorem, intermediatevaluetheorem.

2(Brown; Simpson). Urysohn’slemma, Tietzeextensiontheorem.

3(Rabin). Existenceofalgebraicclosuresofcountablefields.

Workinreversemathematicshasrevealeda strikingfact: mosttheoremsareprovablein RCA0, or equivalent tooneoftheotherfoursystems.

Theorem. Thefollowingareequivalentto WKL0 over RCA0.

1(Brown; Friedman). Heine-Boreltheoremfor [0, 1].

2(Orevkov; ShojiandTanaka). Brouwerfixed-pointtheorem.

3(Friedman, Simpson, andSmith). Primeidealtheorem.

Reversemathematics.

Theorem. Thefollowingareequivalentto ACA0 over RCA0.

1(Friedman). Bolzano-Weierstrasstheorem.

2(Dekker). Existenceofbasesinvectorspaces.

3(Friedman, Simpson, andSmith). Maximalidealtheorem.

Themaximalidealtheoremis stronger thantheprimeidealtheorem.

Question. Arethere irregular principles, lyingoutsidethebigfive?

Remark. Inductionin RCA0 islimitedto IΣ01, i.e., to

(φ(0)∧ (∀x)[φ(x) → φ(x+ 1)]) → (∀x)φ(x)

for Σ01 formulas φ. RCA0 cannotprove IΣ0n for n > 1.

Ramsey’stheorem.

Definition. Let [N]n denotethesetofall n-elementsubsetsof N.

Definition. A k-coloring of [N]n isamap f : [N]n → k = {0, 1, . . . , k− 1}.

RT(n, k). Every k-coloringof [N]n hasaninfinitehomogeneousset.

So RT(1, k) isjustthepigeonholeprinciple.

RT(2, k) assertstheexistenceofinfinitecliquesoranticliques.

.. · · ·

Ramsey’stheorem.

Definition. Let [N]n denotethesetofall n-elementsubsetsof N.

Definition. A k-coloring of [N]n isamap f : [N]n → k = {0, 1, . . . , k− 1}.

RT(n, k). Every k-coloringof [N]n hasaninfinitehomogeneousset.

So RT(1, k) isjustthepigeonholeprinciple.

RT(2, k) assertstheexistenceofinfinitecliquesoranticliques.

.. · · ·

Ramsey’stheorem.

RT(1, k) isprovedin RCA0. For n ⩾ 2, RT(n, k) isprovablein ACA0.

Theorem(Jockusch). For n ⩾ 3, RT(n, k) isequivalentto ACA0.

Howstrongis RT(2, k)? Itcanbeshownthatitisnotprovablein WKL0.

Theorem. Let f beacomputable k-coloringof [N]2.

1(Seetapun). Forany C ≰T ∅, f hasaninfinitehomogeneousset H ≱T C.

2(Cholak, Jockusch, andSlaman). f hasalow2 infinitehomogeneousset.

3(DzhafarovandJockusch). f hastwolow2 infinitehomogeneoussets3 H0 and H1 whosedegreeformaminimalpair.

Thisallowsustobuildamodelof RCA0 + RT(2, k) notcomputing ∅ ′.

Corollary. RT(2, k) doesnotimply ACA0 over RCA0.

Ramsey’stheorem.

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Ramsey’stheorem.

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A zooofcombinatorialprinciples.

Elucidatingthestrengthof RT(2, k) hasgivenrisetoanindustryofresearchintothereversemathematicsofcombinatorialprinciples.

Downey, Hirschfeldt, Lempp, andSolomon; Mileti; Liu. Furtherinvestigationoftheinformationcontentof RT(2, k).

Cholak, Jockusch, andSlaman. StableRamsey’stheorem, cohesiveprinciple.

HirschfeldtandShore. Principlesaboutpartialandlinearorders.

DzhafarovandHirst. PolarizedversionsofRamsey’stheorem.

BovykinandWeiermann; Lerman, Solomon, andTowsner. Tournaments.

Ambos-Spiesetal.; Dzhafarov. Combinatorialprinciplesrelatedtonotionsofalgorithmicrandomness.

A zooofcombinatorialprinciples.

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Beyondcombinatorics: theatomicmodeltheorem.

Hirschfeldt, Shore, andSlamanstudiedmodel-theoretictheoremsconcerningwhenatheoryhasaanatomicmodel, oneassmallaspossible. (Atomicmodelsareinitialobjectsinthecategoryofmodels.)

AMT. Everyatomictheoryhasanatomicmodel.

Therearetwovariants, OPT and AST, whicharespecialcasesof AMT.

Theorem(Hirschfeldt, Shore, andSlaman.)

1. AMT isnotprovablein RCA0, butitisextremelyweak: itisimplied1. over RCA0 byvirtuallyeverycombinatorialprinciplebelow RT(2, k).

2. OPT isequivalenttotheexistenceofhyperimmunesets, i.e., itcanbe2. characterizedintermsofgrowthratesofcomputablefunctions.

3. AST isequivalenttotheexistenceofnoncomputablesets.

Beyondcombinatorics: theatomicmodeltheorem.

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Intersectionprinciples.

A familyofsetsissaidtohavethe finiteintersectionproperty(f.i.p.)iftheintersectionofanyfinitelymanyofitsmembersisnon-empty.

FIP. Everyfamilyofsetshasamaximalsubfamilywithf.i.p.

NIP. Everyfamilyofsetshasamaximaldisjointsubfamily.

Over ZF, theseprinciplesareequivalenttotheaxiomofchoice.

Theorem(DzhafarovandMummert).

1. Over RCA0, NIP isequivalentto ACA0.

2. FIP isnotprovablein RCA0, butithassolutionscomputablefromany2. noncomputablec.e. set, aswellasfromanyCohengenericset.

3. Over RCA0, AMT implies FIP, whichimplies OPT.

Intersectionprinciples.

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Computableanduniformreductions.

Animplication P → Q in RCA0 mayuse P severaltimestoobtain Q, orinvolvenon-uniformdecisionsabouthowtoproceedinaconstruction.

Butinmostcases, theimplicationisactuallya computablereduction.

Example. P = “everyinfinitebinarytreehasaninfinitepath”.Example. Q = “everycommutativeringwithunityhasaprimeideal”.

..Instances.

Solutions

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P

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Q

. ring R. binarytree T.

branch B

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ideal I

Computableanduniformreductions.

Animplication P → Q in RCA0 mayuse P severaltimestoobtain Q, orinvolvenon-uniformdecisionsabouthowtoproceedinaconstruction.

Butinmostcases, theimplicationisactuallya computablereduction.

Example. P = “everyinfinitebinarytreehasaninfinitepath”.Example. Q = “everycommutativeringwithunityhasaprimeideal”.

..Instances.

Solutions

.

P

.

Q

. ring R. binarytree T.

branch B

.

ideal I

Computableanduniformreductions.

Animplication P → Q in RCA0 mayuse P severaltimestoobtain Q, orinvolvenon-uniformdecisionsabouthowtoproceedinaconstruction.

Butinmostcases, theimplicationisactuallya computablereduction.

Example. P = “everyinfinitebinarytreehasaninfinitepath”.Example. Q = “everycommutativeringwithunityhasaprimeideal”.

..Instances.

Solutions

.

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Q

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branch B

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ideal I

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Computableanduniformreductions.

Animplication P → Q in RCA0 mayuse P severaltimestoobtain Q, orinvolvenon-uniformdecisionsabouthowtoproceedinaconstruction.

Butinmostcases, theimplicationisactuallya computablereduction.

Example. P = “everyinfinitebinarytreehasaninfinitepath”.Example. Q = “everycommutativeringwithunityhasaprimeideal”.

..Instances.

Solutions

.

P

.

Q

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branch B

.

ideal I

. Φ

Computableanduniformreductions.

Animplication P → Q in RCA0 mayuse P severaltimestoobtain Q, orinvolvenon-uniformdecisionsabouthowtoproceedinaconstruction.

Butinmostcases, theimplicationisactuallya computablereduction.

Example. P = “everyinfinitebinarytreehasaninfinitepath”.Example. Q = “everycommutativeringwithunityhasaprimeideal”.

..Instances.

Solutions

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P

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branch B

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Ψ

Here, thereductionisalso uniform: Φ and Ψ donotdependon R.

StableRamsey’stheorem.

Twoversionsof RT(2, k) aimedatsimplifyingtheproblem:

SRT(2, k). RT(2, k) for stablecolorings: f(x, y) = f(x, z) foralmostall y, z.

COH. Sequentialformof RT(1, k), butallowingfinitelymanymistakes.

Itisknownthat RT(2, k) ↔ SRT(2, k) + COH over RCA0.

OpenQuestion(Cholak, Jockusch, andSlaman). Does SRT(2, k) implyRT(2, k) instandardmodelsof RCA0? I.e., does SRT(2, k) imply COH?

Asapartialsolution, lendingcredencetoa negative answer:

Theorem(Dzhafarov). COH doesnotcomputablyreduceto SRT(2, k).

Thus, if SRT(2, k) implies COH itisnotviathe typical argument. Theproofisaforcingargument: SRT(2, k) doesnot generically imply COH.

Differentnumbersofcolors.

If j < k, then RT(n, j) istriviallyuniformlyreducibleto RT(n, k), viatheidentityreductions. Over RCA0, RT(n, j) alsoimplies RT(n, k).

Question. If j < k, isthere auniformreductionfrom RT(n, k) to RT(n, j)?

Definition. Twoinstancesof RT(n, j) are uniformlyreducibletoone ifDefinition. thereareeffectiveprocedures Φ and Ψ suchthat:Definition. 1. if f0, f1 : [N]n → j then Φ(f0, f1) = g : [N]n → j;Definition. 2. if G isinfiniteand g-homogeneousthen Ψ(G) = ⟨H0,H1⟩Definition. 2. where Hi isinfiniteand fi-homogeneous.

Theorem(Dorais, Dzhafarov, Hirst, Mileti, andShafer). Iftwoinstancesof RT(n, j) areuniformlyreducibletoone, thensoare ω many.

Bycodingcolorsintermsofmultipleinstances, weobtain:

Corollary. If j < k, then RT(n, k) isnotuniformlyreducibleto RT(n, j).

Proofofthesquashingtheorem.

Fix f0, f1, . . . : [N]n → j, andcomputablereductions Φ,Ψ.

Webuildnewcolorings g0, g1, . . . : [N]n → j soastosatisfy

gi(⃗x) = Φ(fi, gi+1)(⃗x)

forlargeenough x⃗ ∈ [N]n. Thus, g0 ≈ Φ(f0, g1).

Proofofthesquashingtheorem.

Fix f0, f1, . . . : [N]n → j, andcomputablereductions Φ,Ψ.

Webuildnewcolorings g0, g1, . . . : [N]n → j soastosatisfy

gi(⃗x) = Φ(fi, gi+1)(⃗x)

forlargeenough x⃗ ∈ [N]n. Thus, g0 ≈ Φ(f0,Φ(f1, g2)).

Proofofthesquashingtheorem.

Fix f0, f1, . . . : [N]n → j, andcomputablereductions Φ,Ψ.

Webuildnewcolorings g0, g1, . . . : [N]n → j soastosatisfy

gi(⃗x) = Φ(fi, gi+1)(⃗x)

forlargeenough x⃗ ∈ [N]n. Thus, g0 ≈ Φ(f0,Φ(f1,Φ(f2, g3))).

Proofofthesquashingtheorem.

Fix f0, f1, . . . : [N]n → j, andcomputablereductions Φ,Ψ.

Webuildnewcolorings g0, g1, . . . : [N]n → j soastosatisfy

gi(⃗x) = Φ(fi, gi+1)(⃗x)

forlargeenough x⃗ ∈ [N]n. Thus, g0 ≈ Φ(f0,Φ(f1,Φ(f2,Φ(f3, · · · )))).

Proofofthesquashingtheorem.

Fix f0, f1, . . . : [N]n → j, andcomputablereductions Φ,Ψ.

Webuildnewcolorings g0, g1, . . . : [N]n → j soastosatisfy

gi(⃗x) = Φ(fi, gi+1)(⃗x)

forlargeenough x⃗ ∈ [N]n. Thus, g0 ≈ Φ(f0,Φ(f1,Φ(f2,Φ(f3, · · · )))).

..g0. g1. g2. g3.

n3

If Gi ishomogeneousfor gi then Gi − ni ishomogeneousfor Φ(fi, gi+1).

Proofofthesquashingtheorem.

Fix f0, f1, . . . : [N]n → j, andcomputablereductions Φ,Ψ.

Webuildnewcolorings g0, g1, . . . : [N]n → j soastosatisfy

gi(⃗x) = Φ(fi, gi+1)(⃗x)

forlargeenough x⃗ ∈ [N]n. Thus, g0 ≈ Φ(f0,Φ(f1,Φ(f2,Φ(f3, · · · )))).

..g0. g1. g2. g3.

Φ(f0, g1)

.

n1

.

n3

If Gi ishomogeneousfor gi then Gi − ni ishomogeneousfor Φ(fi, gi+1).

Proofofthesquashingtheorem.

Fix f0, f1, . . . : [N]n → j, andcomputablereductions Φ,Ψ.

Webuildnewcolorings g0, g1, . . . : [N]n → j soastosatisfy

gi(⃗x) = Φ(fi, gi+1)(⃗x)

forlargeenough x⃗ ∈ [N]n. Thus, g0 ≈ Φ(f0,Φ(f1,Φ(f2,Φ(f3, · · · )))).

..g0. g1. g2. g3.

Φ(f0, g1)

.

Φ(f1, g2)

.

n1

.

n2

.

n3

If Gi ishomogeneousfor gi then Gi − ni ishomogeneousfor Φ(fi, gi+1).

Proofofthesquashingtheorem.

Fix f0, f1, . . . : [N]n → j, andcomputablereductions Φ,Ψ.

Webuildnewcolorings g0, g1, . . . : [N]n → j soastosatisfy

gi(⃗x) = Φ(fi, gi+1)(⃗x)

forlargeenough x⃗ ∈ [N]n. Thus, g0 ≈ Φ(f0,Φ(f1,Φ(f2,Φ(f3, · · · )))).

..g0. g1. g2. g3.

Φ(f0, g1)

.

Φ(f1, g2)

.

Φ(f2, g3)

.

n1

.

n2

.

n3

Thankyouforyourattention!