(Not) Computing linear orders

Noah Schweber

April 23, 2020

Breaking uniform computations:linear orders with “wild” intervals

Measures of complexity, 1/2By structure we will mean a countable structure in a finitelanguage. A copy of a structure is an isomorphic structure withdomain ω. We identify copies with their atomic diagrams.A few natural ways to compare structures:

I Muchnik (weak, nonuniform) reducibility: A ≤w B iff forevery copy B of B there is a copy A of A with A ≤T B.

I Medvedev (strong, uniform) reducibility: A ≤s B iff there issome e ∈ ω such that ΦB

e∼= A for every B ∼= B.

I Medvedev-mod-parameters: A ≤s/p B iff there is some tuplec ∈ B such that A ≤s (B, c).

Theorem (Kallimullin)

Only the obvious implications hold in general (although in allknown natural examples, ≤w and ≤s/p coincide).

Theorem (Turetsky, Westrick)

“≤s/p” is not transitive in general.

Measures of complexity, 1/2By structure we will mean a countable structure in a finitelanguage. A copy of a structure is an isomorphic structure withdomain ω. We identify copies with their atomic diagrams.A few natural ways to compare structures:

I Muchnik (weak, nonuniform) reducibility: A ≤w B iff forevery copy B of B there is a copy A of A with A ≤T B.

I Medvedev (strong, uniform) reducibility: A ≤s B iff there issome e ∈ ω such that ΦB

e∼= A for every B ∼= B.

I Medvedev-mod-parameters: A ≤s/p B iff there is some tuplec ∈ B such that A ≤s (B, c).

Theorem (Kallimullin)

Only the obvious implications hold in general (although in allknown natural examples, ≤w and ≤s/p coincide).

Theorem (Turetsky, Westrick)

“≤s/p” is not transitive in general.

Measures of complexity, 1/2By structure we will mean a countable structure in a finitelanguage. A copy of a structure is an isomorphic structure withdomain ω. We identify copies with their atomic diagrams.A few natural ways to compare structures:

I Muchnik (weak, nonuniform) reducibility: A ≤w B iff forevery copy B of B there is a copy A of A with A ≤T B.

I Medvedev (strong, uniform) reducibility: A ≤s B iff there issome e ∈ ω such that ΦB

e∼= A for every B ∼= B.

I Medvedev-mod-parameters: A ≤s/p B iff there is some tuplec ∈ B such that A ≤s (B, c).

Theorem (Kallimullin)

Only the obvious implications hold in general (although in allknown natural examples, ≤w and ≤s/p coincide).

Theorem (Turetsky, Westrick)

“≤s/p” is not transitive in general.

Measures of complexity, 1/2By structure we will mean a countable structure in a finitelanguage. A copy of a structure is an isomorphic structure withdomain ω. We identify copies with their atomic diagrams.A few natural ways to compare structures:

I Muchnik (weak, nonuniform) reducibility: A ≤w B iff forevery copy B of B there is a copy A of A with A ≤T B.

I Medvedev (strong, uniform) reducibility: A ≤s B iff there issome e ∈ ω such that ΦB

e∼= A for every B ∼= B.

I Medvedev-mod-parameters: A ≤s/p B iff there is some tuplec ∈ B such that A ≤s (B, c).

Theorem (Kallimullin)

Only the obvious implications hold in general (although in allknown natural examples, ≤w and ≤s/p coincide).

Theorem (Turetsky, Westrick)

“≤s/p” is not transitive in general.

Measures of complexity, 1/2By structure we will mean a countable structure in a finitelanguage. A copy of a structure is an isomorphic structure withdomain ω. We identify copies with their atomic diagrams.A few natural ways to compare structures:

I Muchnik (weak, nonuniform) reducibility: A ≤w B iff forevery copy B of B there is a copy A of A with A ≤T B.

I Medvedev (strong, uniform) reducibility: A ≤s B iff there issome e ∈ ω such that ΦB

e∼= A for every B ∼= B.

I Medvedev-mod-parameters: A ≤s/p B iff there is some tuplec ∈ B such that A ≤s (B, c).

Theorem (Kallimullin)

Only the obvious implications hold in general (although in allknown natural examples, ≤w and ≤s/p coincide).

Theorem (Turetsky, Westrick)

“≤s/p” is not transitive in general.

Measures of complexity, 1/2By structure we will mean a countable structure in a finitelanguage. A copy of a structure is an isomorphic structure withdomain ω. We identify copies with their atomic diagrams.A few natural ways to compare structures:

I Muchnik (weak, nonuniform) reducibility: A ≤w B iff forevery copy B of B there is a copy A of A with A ≤T B.

I Medvedev (strong, uniform) reducibility: A ≤s B iff there issome e ∈ ω such that ΦB

e∼= A for every B ∼= B.

I Medvedev-mod-parameters: A ≤s/p B iff there is some tuplec ∈ B such that A ≤s (B, c).

Theorem (Kallimullin)

Only the obvious implications hold in general (although in allknown natural examples, ≤w and ≤s/p coincide).

Theorem (Turetsky, Westrick)

“≤s/p” is not transitive in general.

Measures of complexity, 2/2Muchnik reducibility is much better understood than Medvedevreducibility.

Theorem (Essentially Sacks)

For countable ordinals α, β (thought of as linear orders) we haveα ≤w β iff α < ωCK

1 (β).

Question (Hamkins, Li)

What does the degree structure Dords of the countable ordinals

modulo ≤s look like?

This is still largely open;

we know (S.) that there is a club ofMedvedev-incomparable ordinals and that under mild assumptionsthere is no embedding of ω1 into Dord

s . But even very concretequestions are still open:

QuestionIs ωCK

1 ≤s ωCK2 ? Do joins exist in Dord

s ?

Things are a bit clearer if we look at uniform hyperarithmeticreducibility, but still largely unknown.

Measures of complexity, 2/2Muchnik reducibility is much better understood than Medvedevreducibility.

Theorem (Essentially Sacks)

For countable ordinals α, β (thought of as linear orders) we haveα ≤w β iff α < ωCK

1 (β).

Question (Hamkins, Li)

What does the degree structure Dords of the countable ordinals

modulo ≤s look like?

This is still largely open; we know (S.) that there is a club ofMedvedev-incomparable ordinals and that under mild assumptionsthere is no embedding of ω1 into Dord

s . But even very concretequestions are still open:

QuestionIs ωCK

1 ≤s ωCK2 ? Do joins exist in Dord

s ?

Things are a bit clearer if we look at uniform hyperarithmeticreducibility, but still largely unknown.

Measures of complexity, 2/2Muchnik reducibility is much better understood than Medvedevreducibility.

Theorem (Essentially Sacks)

For countable ordinals α, β (thought of as linear orders) we haveα ≤w β iff α < ωCK

1 (β).

Question (Hamkins, Li)

What does the degree structure Dords of the countable ordinals

modulo ≤s look like?

This is still largely open; we know (S.) that there is a club ofMedvedev-incomparable ordinals and that under mild assumptionsthere is no embedding of ω1 into Dord

s . But even very concretequestions are still open:

QuestionIs ωCK

1 ≤s ωCK2 ? Do joins exist in Dord

s ?

Things are a bit clearer if we look at uniform hyperarithmeticreducibility, but still largely unknown.

Endpointed intervals

DefinitionA wild interval in a linear order L is an endpointed interval [a, b]Lwith [a, b]L 6≤s L. L is wild if L has a wild interval.

(Obviously impossible for ≤w .)

Since well-orderings comparenicely, ≤s -incomparable ordinals yield wild intervals. Unfortunately,the examples produced are hideously large: images of ω1 underMostowski collapse of “sufficiently large” countable substructuresof the universe.Can we do better?

Endpointed intervals

DefinitionA wild interval in a linear order L is an endpointed interval [a, b]Lwith [a, b]L 6≤s L. L is wild if L has a wild interval.

(Obviously impossible for ≤w .) Since well-orderings comparenicely, ≤s -incomparable ordinals yield wild intervals.

Unfortunately,the examples produced are hideously large: images of ω1 underMostowski collapse of “sufficiently large” countable substructuresof the universe.Can we do better?

Endpointed intervals

DefinitionA wild interval in a linear order L is an endpointed interval [a, b]Lwith [a, b]L 6≤s L. L is wild if L has a wild interval.

(Obviously impossible for ≤w .) Since well-orderings comparenicely, ≤s -incomparable ordinals yield wild intervals. Unfortunately,the examples produced are hideously large: images of ω1 underMostowski collapse of “sufficiently large” countable substructuresof the universe.Can we do better?

A simple wild interval

Theorem (Knight, Soskova)

There wild linear orders low in the arithmetical hierarchy.

(Sacks: impossible for ordinals.) Key observation:

A ≤s B =⇒ ΣcnTh(A) ≤e Σc

nTh(B).

Now look for “shapeable” linear orders with well-understoodΣcn-theories for some rich enough n: shuffle sums.

DefinitionFor X ⊆ ω + 1, Shuff (X ) is the unique-up-to-isomorphism linearorder of the form

∑q∈Q Lq where each Lq has ordertype n for

some n ∈ X and {q ∈ Q : |Lq| = n} is dense for n ∈ X .

Knight-Soskova use

Shuff (X ∪ {ω}) + 1 + Shuff (ω + 1)

for X 6∈ Σ03.

A simple wild interval

Theorem (Knight, Soskova)

There wild linear orders low in the arithmetical hierarchy.

(Sacks: impossible for ordinals.) Key observation:

A ≤s B =⇒ ΣcnTh(A) ≤e Σc

nTh(B).

Now look for “shapeable” linear orders with well-understoodΣcn-theories for some rich enough n: shuffle sums.

DefinitionFor X ⊆ ω + 1, Shuff (X ) is the unique-up-to-isomorphism linearorder of the form

∑q∈Q Lq where each Lq has ordertype n for

some n ∈ X and {q ∈ Q : |Lq| = n} is dense for n ∈ X .

Knight-Soskova use

Shuff (X ∪ {ω}) + 1 + Shuff (ω + 1)

for X 6∈ Σ03.

A simple wild interval

Theorem (Knight, Soskova)

There wild linear orders low in the arithmetical hierarchy.

(Sacks: impossible for ordinals.) Key observation:

A ≤s B =⇒ ΣcnTh(A) ≤e Σc

nTh(B).

Now look for “shapeable” linear orders with well-understoodΣcn-theories for some rich enough n: shuffle sums.

DefinitionFor X ⊆ ω + 1, Shuff (X ) is the unique-up-to-isomorphism linearorder of the form

∑q∈Q Lq where each Lq has ordertype n for

some n ∈ X and {q ∈ Q : |Lq| = n} is dense for n ∈ X .

Knight-Soskova use

Shuff (X ∪ {ω}) + 1 + Shuff (ω + 1)

for X 6∈ Σ03.

A simple wild interval

Theorem (Knight, Soskova)

There wild linear orders low in the arithmetical hierarchy.

(Sacks: impossible for ordinals.) Key observation:

A ≤s B =⇒ ΣcnTh(A) ≤e Σc

nTh(B).

Now look for “shapeable” linear orders with well-understoodΣcn-theories for some rich enough n: shuffle sums.

DefinitionFor X ⊆ ω + 1, Shuff (X ) is the unique-up-to-isomorphism linearorder of the form

∑q∈Q Lq where each Lq has ordertype n for

some n ∈ X and {q ∈ Q : |Lq| = n} is dense for n ∈ X .

Knight-Soskova use

Shuff (X ∪ {ω}) + 1 + Shuff (ω + 1)

for X 6∈ Σ03.

A simple wild interval

Theorem (Knight, Soskova)

There wild linear orders low in the arithmetical hierarchy.

(Sacks: impossible for ordinals.) Key observation:

A ≤s B =⇒ ΣcnTh(A) ≤e Σc

nTh(B).

Now look for “shapeable” linear orders with well-understoodΣcn-theories for some rich enough n: shuffle sums.

DefinitionFor X ⊆ ω + 1, Shuff (X ) is the unique-up-to-isomorphism linearorder of the form

∑q∈Q Lq where each Lq has ordertype n for

some n ∈ X and {q ∈ Q : |Lq| = n} is dense for n ∈ X .

Knight-Soskova use

Shuff (X ∪ {ω}) + 1 + Shuff (ω + 1)

for X 6∈ Σ03.

Beyond shuffle sums

Can we find more examples?

QuestionIs there a scattered wild linear order low in the arithmeticalhierarchy?

QuestionCan a linear order be “totally wild” (whatever that means)?

Former appears to point towards ordinals; latter interesting on itsown.

Theorem (S.)

Yes to both ... unhelpfully.

Both arguments via forcing. Nice feature: “coarse” argumentrelativizes to generally uniform reducibility notions (although weeventually need large cardinals), while Σc

n-analysis-approachdoesn’t seem to.

Beyond shuffle sums

Can we find more examples?

QuestionIs there a scattered wild linear order low in the arithmeticalhierarchy?

QuestionCan a linear order be “totally wild” (whatever that means)?

Former appears to point towards ordinals; latter interesting on itsown.

Theorem (S.)

Yes to both ... unhelpfully.

Both arguments via forcing. Nice feature: “coarse” argumentrelativizes to generally uniform reducibility notions (although weeventually need large cardinals), while Σc

n-analysis-approachdoesn’t seem to.

Beyond shuffle sums

Can we find more examples?

QuestionIs there a scattered wild linear order low in the arithmeticalhierarchy?

QuestionCan a linear order be “totally wild” (whatever that means)?

Former appears to point towards ordinals; latter interesting on itsown.

Theorem (S.)

Yes to both ... unhelpfully.

Both arguments via forcing. Nice feature: “coarse” argumentrelativizes to generally uniform reducibility notions (although weeventually need large cardinals), while Σc

n-analysis-approachdoesn’t seem to.

A scattered wild linear orderWe code reals into linear orders, then build a Z-sum withseparators out of a Cohen array.

I For X ⊆ ω, n ∈ ω let

A(X , i) =

{ζ + i + 2 + ζ if i ∈ X ,

ζ + i + 3 + ζ if i 6∈ X

and set A(X ) = ζ +∑

i∈ω A(X , i) + ζ.I For X = (Xz)z∈Z a doubly-infinite sequence of reals, let

S(X ) =∑z∈Z

[1 + A(Xz)]

= ...+ 1 + A(X−1) + 1 + A(X0) + 1 + A(X1) + 1 + ...

S(X ) is scattered. We’ll look at S(X ) for X sufficiently (?)generic:

DefinitionPsca is the set of arrays p ∈ (2<ω)Z with pz = ∅ for cofinitely manyz ∈ Z. (This is just Cohen forcing rephrased.)

A scattered wild linear orderWe code reals into linear orders, then build a Z-sum withseparators out of a Cohen array.

I For X ⊆ ω, n ∈ ω let

A(X , i) =

{ζ + i + 2 + ζ if i ∈ X ,

ζ + i + 3 + ζ if i 6∈ X

and set A(X ) = ζ +∑

i∈ω A(X , i) + ζ.I For X = (Xz)z∈Z a doubly-infinite sequence of reals, let

S(X ) =∑z∈Z

[1 + A(Xz)]

= ...+ 1 + A(X−1) + 1 + A(X0) + 1 + A(X1) + 1 + ...

S(X ) is scattered. We’ll look at S(X ) for X sufficiently (?)generic:

DefinitionPsca is the set of arrays p ∈ (2<ω)Z with pz = ∅ for cofinitely manyz ∈ Z. (This is just Cohen forcing rephrased.)

A scattered wild linear orderWe code reals into linear orders, then build a Z-sum withseparators out of a Cohen array.I For X ⊆ ω, n ∈ ω let

A(X , i) =

{ζ + i + 2 + ζ if i ∈ X ,

ζ + i + 3 + ζ if i 6∈ X

and set A(X ) = ζ +∑

i∈ω A(X , i) + ζ.

I For X = (Xz)z∈Z a doubly-infinite sequence of reals, let

S(X ) =∑z∈Z

[1 + A(Xz)]

= ...+ 1 + A(X−1) + 1 + A(X0) + 1 + A(X1) + 1 + ...

S(X ) is scattered. We’ll look at S(X ) for X sufficiently (?)generic:

DefinitionPsca is the set of arrays p ∈ (2<ω)Z with pz = ∅ for cofinitely manyz ∈ Z. (This is just Cohen forcing rephrased.)

A scattered wild linear orderWe code reals into linear orders, then build a Z-sum withseparators out of a Cohen array.I For X ⊆ ω, n ∈ ω let

A(X , i) =

{ζ + i + 2 + ζ if i ∈ X ,

ζ + i + 3 + ζ if i 6∈ X

and set A(X ) = ζ +∑

i∈ω A(X , i) + ζ.I For X = (Xz)z∈Z a doubly-infinite sequence of reals, let

S(X ) =∑z∈Z

[1 + A(Xz)]

= ...+ 1 + A(X−1) + 1 + A(X0) + 1 + A(X1) + 1 + ...

S(X ) is scattered. We’ll look at S(X ) for X sufficiently (?)generic:

DefinitionPsca is the set of arrays p ∈ (2<ω)Z with pz = ∅ for cofinitely manyz ∈ Z. (This is just Cohen forcing rephrased.)

A scattered wild linear orderWe code reals into linear orders, then build a Z-sum withseparators out of a Cohen array.I For X ⊆ ω, n ∈ ω let

A(X , i) =

{ζ + i + 2 + ζ if i ∈ X ,

ζ + i + 3 + ζ if i 6∈ X

and set A(X ) = ζ +∑

i∈ω A(X , i) + ζ.I For X = (Xz)z∈Z a doubly-infinite sequence of reals, let

S(X ) =∑z∈Z

[1 + A(Xz)]

= ...+ 1 + A(X−1) + 1 + A(X0) + 1 + A(X1) + 1 + ...

S(X ) is scattered. We’ll look at S(X ) for X sufficiently (?)generic:

DefinitionPsca is the set of arrays p ∈ (2<ω)Z with pz = ∅ for cofinitely manyz ∈ Z. (This is just Cohen forcing rephrased.)

A scattered wild linear orderWe code reals into linear orders, then build a Z-sum withseparators out of a Cohen array.I For X ⊆ ω, n ∈ ω let

A(X , i) =

{ζ + i + 2 + ζ if i ∈ X ,

ζ + i + 3 + ζ if i 6∈ X

and set A(X ) = ζ +∑

i∈ω A(X , i) + ζ.I For X = (Xz)z∈Z a doubly-infinite sequence of reals, let

S(X ) =∑z∈Z

[1 + A(Xz)]

= ...+ 1 + A(X−1) + 1 + A(X0) + 1 + A(X1) + 1 + ...

S(X ) is scattered. We’ll look at S(X ) for X sufficiently (?)generic:

DefinitionPsca is the set of arrays p ∈ (2<ω)Z with pz = ∅ for cofinitely manyz ∈ Z. (This is just Cohen forcing rephrased.)

S(X ) = ...+ 1 + A(X−1) + 1 + A(X0) + 1 + A(X1) + 1 + ...

Let IXz be the (1 + A(Xz) + 1)-part of S(X ); under mild genericity,this is unique.

TheoremSuppose X is sufficiently (?) Psca-generic. Then for each z ∈ Z wehave Iz 6≤s S(X ).

Idea:

I Shifts of Z yield automorphisms of Psca which preserve theisomorphism type of the linear order built.

I Finite support (= cofinitely many partial reals empty at eachstage) lets us build divergent extensions of conditions.

ConventionLook at z = 0 for simplicity, let G be (name for) Psca-generic filter,and assume (pz)z∈Z = p ∈ Psca forces Φ−e : S(G) ≥s IG0 . (?)

S(X ) = ...+ 1 + A(X−1) + 1 + A(X0) + 1 + A(X1) + 1 + ...

Let IXz be the (1 + A(Xz) + 1)-part of S(X ); under mild genericity,this is unique.

TheoremSuppose X is sufficiently (?) Psca-generic. Then for each z ∈ Z wehave Iz 6≤s S(X ).

Idea:

I Shifts of Z yield automorphisms of Psca which preserve theisomorphism type of the linear order built.

I Finite support (= cofinitely many partial reals empty at eachstage) lets us build divergent extensions of conditions.

ConventionLook at z = 0 for simplicity, let G be (name for) Psca-generic filter,and assume (pz)z∈Z = p ∈ Psca forces Φ−e : S(G) ≥s IG0 . (?)

S(X ) = ...+ 1 + A(X−1) + 1 + A(X0) + 1 + A(X1) + 1 + ...

Let IXz be the (1 + A(Xz) + 1)-part of S(X ); under mild genericity,this is unique.

TheoremSuppose X is sufficiently (?) Psca-generic. Then for each z ∈ Z wehave Iz 6≤s S(X ).

Idea:

I Shifts of Z yield automorphisms of Psca which preserve theisomorphism type of the linear order built.

I Finite support (= cofinitely many partial reals empty at eachstage) lets us build divergent extensions of conditions.

ConventionLook at z = 0 for simplicity, let G be (name for) Psca-generic filter,and assume (pz)z∈Z = p ∈ Psca forces Φ−e : S(G) ≥s IG0 . (?)

Shift and split

Set σ : z 7→ z + k for some “big enough” k (and conflate σobvious extension to conditions/filters),

and note that if Y0,Y1differ by a shift then S(Y0) ∼= S(Y1).Choose extensions pleft , pright ≤ p such that σ(pleft) = pright and

pleft0 (n) ↓6= pright

0 (n) for some “big enough” n.Picking generics G left 3 pleft ,Gright 3 pright with σ(G left) = Grightwe have:

I IGleft

0 6∼= IGright

0 ,

I S(G left) ∼= S(Gright) (call this “S”),

I ΦS(G left)e

∼= IGleft

0 and ΦS(Gright)e

∼= IGright

0 , but

I Φe is isomorphism-invariant on copies of S

Shift and split

Set σ : z 7→ z + k for some “big enough” k (and conflate σobvious extension to conditions/filters), and note that if Y0,Y1differ by a shift then S(Y0) ∼= S(Y1).

Choose extensions pleft , pright ≤ p such that σ(pleft) = pright and

pleft0 (n) ↓6= pright

0 (n) for some “big enough” n.Picking generics G left 3 pleft ,Gright 3 pright with σ(G left) = Grightwe have:

I IGleft

0 6∼= IGright

0 ,

I S(G left) ∼= S(Gright) (call this “S”),

I ΦS(G left)e

∼= IGleft

0 and ΦS(Gright)e

∼= IGright

0 , but

I Φe is isomorphism-invariant on copies of S

Shift and split

Set σ : z 7→ z + k for some “big enough” k (and conflate σobvious extension to conditions/filters), and note that if Y0,Y1differ by a shift then S(Y0) ∼= S(Y1).Choose extensions pleft , pright ≤ p such that σ(pleft) = pright and

pleft0 (n) ↓6= pright

0 (n) for some “big enough” n.

Picking generics G left 3 pleft ,Gright 3 pright with σ(G left) = Grightwe have:

I IGleft

0 6∼= IGright

0 ,

I S(G left) ∼= S(Gright) (call this “S”),

I ΦS(G left)e

∼= IGleft

0 and ΦS(Gright)e

∼= IGright

0 , but

I Φe is isomorphism-invariant on copies of S

Shift and split

Set σ : z 7→ z + k for some “big enough” k (and conflate σobvious extension to conditions/filters), and note that if Y0,Y1differ by a shift then S(Y0) ∼= S(Y1).Choose extensions pleft , pright ≤ p such that σ(pleft) = pright and

pleft0 (n) ↓6= pright

0 (n) for some “big enough” n.Picking generics G left 3 pleft ,Gright 3 pright with σ(G left) = Grightwe have:

I IGleft

0 6∼= IGright

0 ,

I S(G left) ∼= S(Gright) (call this “S”),

I ΦS(G left)e

∼= IGleft

0 and ΦS(Gright)e

∼= IGright

0 , but

I Φe is isomorphism-invariant on copies of S

Shift and split

Set σ : z 7→ z + k for some “big enough” k (and conflate σobvious extension to conditions/filters), and note that if Y0,Y1differ by a shift then S(Y0) ∼= S(Y1).Choose extensions pleft , pright ≤ p such that σ(pleft) = pright and

pleft0 (n) ↓6= pright

0 (n) for some “big enough” n.Picking generics G left 3 pleft ,Gright 3 pright with σ(G left) = Grightwe have:

I IGleft

0 6∼= IGright

0 ,

I S(G left) ∼= S(Gright) (call this “S”),

I ΦS(G left)e

∼= IGleft

0 and ΦS(Gright)e

∼= IGright

0 , but

I Φe is isomorphism-invariant on copies of S

Shift and split

Set σ : z 7→ z + k for some “big enough” k (and conflate σobvious extension to conditions/filters), and note that if Y0,Y1differ by a shift then S(Y0) ∼= S(Y1).Choose extensions pleft , pright ≤ p such that σ(pleft) = pright and

pleft0 (n) ↓6= pright

0 (n) for some “big enough” n.Picking generics G left 3 pleft ,Gright 3 pright with σ(G left) = Grightwe have:

I IGleft

0 6∼= IGright

0 ,

I S(G left) ∼= S(Gright) (call this “S”),

I ΦS(G left)e

∼= IGleft

0 and ΦS(Gright)e

∼= IGright

0 , but

I Φe is isomorphism-invariant on copies of S

Shift and split

Set σ : z 7→ z + k for some “big enough” k (and conflate σobvious extension to conditions/filters), and note that if Y0,Y1differ by a shift then S(Y0) ∼= S(Y1).Choose extensions pleft , pright ≤ p such that σ(pleft) = pright and

pleft0 (n) ↓6= pright

0 (n) for some “big enough” n.Picking generics G left 3 pleft ,Gright 3 pright with σ(G left) = Grightwe have:

I IGleft

0 6∼= IGright

0 ,

I S(G left) ∼= S(Gright) (call this “S”),

I ΦS(G left)e

∼= IGleft

0 and ΦS(Gright)e

∼= IGright

0 , but

I Φe is isomorphism-invariant on copies of S

Shift and split

Set σ : z 7→ z + k for some “big enough” k (and conflate σobvious extension to conditions/filters), and note that if Y0,Y1differ by a shift then S(Y0) ∼= S(Y1).Choose extensions pleft , pright ≤ p such that σ(pleft) = pright and

pleft0 (n) ↓6= pright

0 (n) for some “big enough” n.Picking generics G left 3 pleft ,Gright 3 pright with σ(G left) = Grightwe have:

I IGleft

0 6∼= IGright

0 ,

I S(G left) ∼= S(Gright) (call this “S”),

I ΦS(G left)e

∼= IGleft

0 and ΦS(Gright)e

∼= IGright

0 , but

I Φe is isomorphism-invariant on copies of S

Even more wild intervals

Next, we’ll build a linear order which has as many wild intervals aspossible - even relative to parameters - in a precise sense.

The extreme example

DefinitionA non-computably-presentable linear order L is a thicket if forevery finite tuple c ∈ L and every infinite [a, b]L we have(L, c) ≥s [a, b]L iff there are ca, cb ∈ c finitely far from a, brespectively.

Easy to see that:

I Every thicket has the form ∑i∈η̂

Bi

for η̂ ∈ {η, 1 + η, η + 1, 1 + η + 1} andBi ∈ ω̂ := ω ∪ {ω, ω∗, ζ}.

I Thickets are basically interwoven nowhere-dense sets: for eachnontrivial interval J ⊆ η̂, the set{q ∈ J : Bq appears nowhere densely in J} is dense in J.

The extreme example

DefinitionA non-computably-presentable linear order L is a thicket if forevery finite tuple c ∈ L and every infinite [a, b]L we have(L, c) ≥s [a, b]L iff there are ca, cb ∈ c finitely far from a, brespectively.

Easy to see that:

I Every thicket has the form ∑i∈η̂

Bi

for η̂ ∈ {η, 1 + η, η + 1, 1 + η + 1} andBi ∈ ω̂ := ω ∪ {ω, ω∗, ζ}.

I Thickets are basically interwoven nowhere-dense sets: for eachnontrivial interval J ⊆ η̂, the set{q ∈ J : Bq appears nowhere densely in J} is dense in J.

The extreme example

DefinitionA non-computably-presentable linear order L is a thicket if forevery finite tuple c ∈ L and every infinite [a, b]L we have(L, c) ≥s [a, b]L iff there are ca, cb ∈ c finitely far from a, brespectively.

Easy to see that:

I Every thicket has the form ∑i∈η̂

Bi

for η̂ ∈ {η, 1 + η, η + 1, 1 + η + 1} andBi ∈ ω̂ := ω ∪ {ω, ω∗, ζ}.

I Thickets are basically interwoven nowhere-dense sets: for eachnontrivial interval J ⊆ η̂, the set{q ∈ J : Bq appears nowhere densely in J} is dense in J.

Co-shuffle sums

Simplest examples of thickets happen to be co-shuffle-sums:

DefinitionA co-shuffle sum is a linear order of the form∑

q∈Qnq

with nq ∈ ω such that for each i ∈ ω the set {q ∈ Q : nq = i} isnowhere dense.

Contra shuffle-sums, there are many co-shuffle sums up toisomorphism and their Σc

n-theories look complicated.

As before, we’ll build a co-shuffle sum from anappropriately-generic family of reals, then use asymmetry-and-splitting argument to kill off potential Medvedevreductions. We can accommodate parameters now since η issufficiently homogeneous (couldn’t do that above).

Co-shuffle sums

Simplest examples of thickets happen to be co-shuffle-sums:

DefinitionA co-shuffle sum is a linear order of the form∑

q∈Qnq

with nq ∈ ω such that for each i ∈ ω the set {q ∈ Q : nq = i} isnowhere dense.

Contra shuffle-sums, there are many co-shuffle sums up toisomorphism and their Σc

n-theories look complicated.As before, we’ll build a co-shuffle sum from anappropriately-generic family of reals, then use asymmetry-and-splitting argument to kill off potential Medvedevreductions. We can accommodate parameters now since η issufficiently homogeneous (couldn’t do that above).

The forcing

Vaguely Mathias-flavored

Conditions: π = 〈a,B〉 where:

I a : Q→ N partial finite,

I B is a finite partial map from closed intervals in R withdistinct finite algebraic irrational endpoints to N,

I for each 〈[x , y ], n〉 ∈ B and q ∈ dom(a), if x < q < y thena(q) 6= n.

Order conditions as with Mathias.Using irrational endpoints for the prohibitions in B isn’t necessarybut gives us “sufficiently homogeneous” intervals for convenience.

The forcing

Vaguely Mathias-flavoredConditions: π = 〈a,B〉 where:

I a : Q→ N partial finite,

I B is a finite partial map from closed intervals in R withdistinct finite algebraic irrational endpoints to N,

I for each 〈[x , y ], n〉 ∈ B and q ∈ dom(a), if x < q < y thena(q) 6= n.

Order conditions as with Mathias.

Using irrational endpoints for the prohibitions in B isn’t necessarybut gives us “sufficiently homogeneous” intervals for convenience.

The forcing

Vaguely Mathias-flavoredConditions: π = 〈a,B〉 where:

I a : Q→ N partial finite,

I B is a finite partial map from closed intervals in R withdistinct finite algebraic irrational endpoints to N,

I for each 〈[x , y ], n〉 ∈ B and q ∈ dom(a), if x < q < y thena(q) 6= n.

Order conditions as with Mathias.Using irrational endpoints for the prohibitions in B isn’t necessarybut gives us “sufficiently homogeneous” intervals for convenience.

Orderings from filters

A barely generic filter G yields a co-shuffle sum L(G )

:

I The domain of L(G ) is the set

{〈q, k〉 ∈ Q× N : k < G (q)}.

I The ordering on L(G ) is the lexicographic ordering

〈q, k〉 ≤L(G) 〈q′, k ′〉 ⇐⇒ (q < q′) ∨ (q = q′ ∧ k ≤ k ′).

For rationals q0 < q1 we write “[q0, q1]G” for the sub-order ofL(G ) consisting of all points with left coordinate q satisfyingq0 ≤ q ≤ q1.For c ∈ Q and G sufficiently P-generic, we let 〈L(G ); c〉 be theexpansion of L(G ) by constants naming each element with leftcoordinate in c.

Orderings from filters

A barely generic filter G yields a co-shuffle sum L(G ):

I The domain of L(G ) is the set

{〈q, k〉 ∈ Q× N : k < G (q)}.

I The ordering on L(G ) is the lexicographic ordering

〈q, k〉 ≤L(G) 〈q′, k ′〉 ⇐⇒ (q < q′) ∨ (q = q′ ∧ k ≤ k ′).

For rationals q0 < q1 we write “[q0, q1]G” for the sub-order ofL(G ) consisting of all points with left coordinate q satisfyingq0 ≤ q ≤ q1.For c ∈ Q and G sufficiently P-generic, we let 〈L(G ); c〉 be theexpansion of L(G ) by constants naming each element with leftcoordinate in c.

Orderings from filters

A barely generic filter G yields a co-shuffle sum L(G ):

I The domain of L(G ) is the set

{〈q, k〉 ∈ Q× N : k < G (q)}.

I The ordering on L(G ) is the lexicographic ordering

〈q, k〉 ≤L(G) 〈q′, k ′〉 ⇐⇒ (q < q′) ∨ (q = q′ ∧ k ≤ k ′).

For rationals q0 < q1 we write “[q0, q1]G” for the sub-order ofL(G ) consisting of all points with left coordinate q satisfyingq0 ≤ q ≤ q1.For c ∈ Q and G sufficiently P-generic, we let 〈L(G ); c〉 be theexpansion of L(G ) by constants naming each element with leftcoordinate in c.

Orderings from filters

A barely generic filter G yields a co-shuffle sum L(G ):

I The domain of L(G ) is the set

{〈q, k〉 ∈ Q× N : k < G (q)}.

I The ordering on L(G ) is the lexicographic ordering

〈q, k〉 ≤L(G) 〈q′, k ′〉 ⇐⇒ (q < q′) ∨ (q = q′ ∧ k ≤ k ′).

For rationals q0 < q1 we write “[q0, q1]G” for the sub-order ofL(G ) consisting of all points with left coordinate q satisfyingq0 ≤ q ≤ q1.

For c ∈ Q and G sufficiently P-generic, we let 〈L(G ); c〉 be theexpansion of L(G ) by constants naming each element with leftcoordinate in c.

Orderings from filters

A barely generic filter G yields a co-shuffle sum L(G ):

I The domain of L(G ) is the set

{〈q, k〉 ∈ Q× N : k < G (q)}.

I The ordering on L(G ) is the lexicographic ordering

〈q, k〉 ≤L(G) 〈q′, k ′〉 ⇐⇒ (q < q′) ∨ (q = q′ ∧ k ≤ k ′).

For rationals q0 < q1 we write “[q0, q1]G” for the sub-order ofL(G ) consisting of all points with left coordinate q satisfyingq0 ≤ q ≤ q1.For c ∈ Q and G sufficiently P-generic, we let 〈L(G ); c〉 be theexpansion of L(G ) by constants naming each element with leftcoordinate in c.

Automorphisms

It will be enough to show that for every tuple c of rationals andevery pair of rationals p < q not both in c , we have

〈L(G ); c〉 6≥s [p, q]G .

DefinitionBy automorphism we’ll mean an automorphism of the linear orderR which sends rationals to rationals and irrationals to irrationalswhose restriction to the rationals is computable.

Automorphisms preserve genericity, and extend naturally to mapson conditions and filters (with which we conflate them).Automorphic filters yield isomorphic orderings.

Automorphisms

It will be enough to show that for every tuple c of rationals andevery pair of rationals p < q not both in c , we have

〈L(G ); c〉 6≥s [p, q]G .

DefinitionBy automorphism we’ll mean an automorphism of the linear orderR which sends rationals to rationals and irrationals to irrationalswhose restriction to the rationals is computable.

Automorphisms preserve genericity, and extend naturally to mapson conditions and filters (with which we conflate them).Automorphic filters yield isomorphic orderings.

Automorphisms

It will be enough to show that for every tuple c of rationals andevery pair of rationals p < q not both in c , we have

〈L(G ); c〉 6≥s [p, q]G .

DefinitionBy automorphism we’ll mean an automorphism of the linear orderR which sends rationals to rationals and irrationals to irrationalswhose restriction to the rationals is computable.

Automorphisms preserve genericity, and extend naturally to mapson conditions and filters (with which we conflate them).Automorphic filters yield isomorphic orderings.

Some structure

DefinitionFix a condition π = 〈a,B〉 ∈ P, irrationals x < y , and a rationalq ∈ [x , y ]. The interval [x , y ] is 〈π, q〉-homogeneous iff

I dom(a) ∩ [x , y ] = {q}, and

I whenever 〈u, v , n〉 ∈ B we have either [u, v ] ∩ [x , y ] = ∅ oru < x < y < v .

LemmaFor every condition π = 〈a,B〉 and every rational q ∈ dom(a)there is a 〈π, q〉-homogeneous interval.

Useful here: irrationality of prohibitions implies that no element ofdom(a) is “critical.”

Some structure

DefinitionFix a condition π = 〈a,B〉 ∈ P, irrationals x < y , and a rationalq ∈ [x , y ]. The interval [x , y ] is 〈π, q〉-homogeneous iff

I dom(a) ∩ [x , y ] = {q}, and

I whenever 〈u, v , n〉 ∈ B we have either [u, v ] ∩ [x , y ] = ∅ oru < x < y < v .

LemmaFor every condition π = 〈a,B〉 and every rational q ∈ dom(a)there is a 〈π, q〉-homogeneous interval.

Useful here: irrationality of prohibitions implies that no element ofdom(a) is “critical.”

Some structure

DefinitionFix a condition π = 〈a,B〉 ∈ P, irrationals x < y , and a rationalq ∈ [x , y ]. The interval [x , y ] is 〈π, q〉-homogeneous iff

I dom(a) ∩ [x , y ] = {q}, and

I whenever 〈u, v , n〉 ∈ B we have either [u, v ] ∩ [x , y ] = ∅ oru < x < y < v .

LemmaFor every condition π = 〈a,B〉 and every rational q ∈ dom(a)there is a 〈π, q〉-homogeneous interval.

Useful here: irrationality of prohibitions implies that no element ofdom(a) is “critical.”

More shifting and splitting, 1/3

TheoremIf G is sufficiently (?) generic, then L(G ) is a thicket.

For w ∈ I ∪ N and π a condition, say w is π-huge if for each〈q, n〉 ∈ a and each 〈x , y , k〉 ∈ B we have w > q, n, x , y , k .Fix sufficiently (?) generic G and π = 〈a,B〉 ∈ G withq0, c ∈ dom(a) such that

π ΦGe : 〈L(G); c〉 ≥s [q0, q1]G.

Have the following:

I an s ∈ I with s > max(dom(a));

I a positive natural N > max(ran(a) ∪ ran(B));

I a t ∈ Q and r0, r1 ∈ I such that q0 < t < r0 < r1 [u, r1] is(π, q0)-homogeneous (which exists by Lemma 0.19); and

I an automorphism α which is the identity outside (u, r1) andhas α(r0) < q0 (which exists by the homogeneity of Q).

More shifting and splitting, 1/3

TheoremIf G is sufficiently (?) generic, then L(G ) is a thicket.

For w ∈ I ∪ N and π a condition, say w is π-huge if for each〈q, n〉 ∈ a and each 〈x , y , k〉 ∈ B we have w > q, n, x , y , k .Fix sufficiently (?) generic G and π = 〈a,B〉 ∈ G withq0, c ∈ dom(a) such that

π ΦGe : 〈L(G); c〉 ≥s [q0, q1]G.

Have the following:

I an s ∈ I with s > max(dom(a));

I a positive natural N > max(ran(a) ∪ ran(B));

I a t ∈ Q and r0, r1 ∈ I such that q0 < t < r0 < r1 [u, r1] is(π, q0)-homogeneous (which exists by Lemma 0.19); and

I an automorphism α which is the identity outside (u, r1) andhas α(r0) < q0 (which exists by the homogeneity of Q).

More shifting and splitting, 1/3

TheoremIf G is sufficiently (?) generic, then L(G ) is a thicket.

For w ∈ I ∪ N and π a condition, say w is π-huge if for each〈q, n〉 ∈ a and each 〈x , y , k〉 ∈ B we have w > q, n, x , y , k .

Fix sufficiently (?) generic G and π = 〈a,B〉 ∈ G withq0, c ∈ dom(a) such that

π ΦGe : 〈L(G); c〉 ≥s [q0, q1]G.

Have the following:

I an s ∈ I with s > max(dom(a));

I a positive natural N > max(ran(a) ∪ ran(B));

I a t ∈ Q and r0, r1 ∈ I such that q0 < t < r0 < r1 [u, r1] is(π, q0)-homogeneous (which exists by Lemma 0.19); and

I an automorphism α which is the identity outside (u, r1) andhas α(r0) < q0 (which exists by the homogeneity of Q).

More shifting and splitting, 1/3

TheoremIf G is sufficiently (?) generic, then L(G ) is a thicket.

For w ∈ I ∪ N and π a condition, say w is π-huge if for each〈q, n〉 ∈ a and each 〈x , y , k〉 ∈ B we have w > q, n, x , y , k .Fix sufficiently (?) generic G and π = 〈a,B〉 ∈ G withq0, c ∈ dom(a) such that

π ΦGe : 〈L(G); c〉 ≥s [q0, q1]G.

Have the following:

I an s ∈ I with s > max(dom(a));

I a positive natural N > max(ran(a) ∪ ran(B));

I a t ∈ Q and r0, r1 ∈ I such that q0 < t < r0 < r1 [u, r1] is(π, q0)-homogeneous (which exists by Lemma 0.19); and

I an automorphism α which is the identity outside (u, r1) andhas α(r0) < q0 (which exists by the homogeneity of Q).

More shifting and splitting, 1/3

TheoremIf G is sufficiently (?) generic, then L(G ) is a thicket.

For w ∈ I ∪ N and π a condition, say w is π-huge if for each〈q, n〉 ∈ a and each 〈x , y , k〉 ∈ B we have w > q, n, x , y , k .Fix sufficiently (?) generic G and π = 〈a,B〉 ∈ G withq0, c ∈ dom(a) such that

π ΦGe : 〈L(G); c〉 ≥s [q0, q1]G.

Have the following:

I an s ∈ I with s > max(dom(a));

I a positive natural N > max(ran(a) ∪ ran(B));

I a t ∈ Q and r0, r1 ∈ I such that q0 < t < r0 < r1 [u, r1] is(π, q0)-homogeneous (which exists by Lemma 0.19); and

I an automorphism α which is the identity outside (u, r1) andhas α(r0) < q0 (which exists by the homogeneity of Q).

More shifting and splitting, 1/3

TheoremIf G is sufficiently (?) generic, then L(G ) is a thicket.

For w ∈ I ∪ N and π a condition, say w is π-huge if for each〈q, n〉 ∈ a and each 〈x , y , k〉 ∈ B we have w > q, n, x , y , k .Fix sufficiently (?) generic G and π = 〈a,B〉 ∈ G withq0, c ∈ dom(a) such that

π ΦGe : 〈L(G); c〉 ≥s [q0, q1]G.

Have the following:

I an s ∈ I with s > max(dom(a));

I a positive natural N > max(ran(a) ∪ ran(B));

I a t ∈ Q and r0, r1 ∈ I such that q0 < t < r0 < r1 [u, r1] is(π, q0)-homogeneous (which exists by Lemma 0.19); and

I an automorphism α which is the identity outside (u, r1) andhas α(r0) < q0 (which exists by the homogeneity of Q).

More shifting and splitting, 1/3

TheoremIf G is sufficiently (?) generic, then L(G ) is a thicket.

For w ∈ I ∪ N and π a condition, say w is π-huge if for each〈q, n〉 ∈ a and each 〈x , y , k〉 ∈ B we have w > q, n, x , y , k .Fix sufficiently (?) generic G and π = 〈a,B〉 ∈ G withq0, c ∈ dom(a) such that

π ΦGe : 〈L(G); c〉 ≥s [q0, q1]G.

Have the following:

I an s ∈ I with s > max(dom(a));

I a positive natural N > max(ran(a) ∪ ran(B));

I a t ∈ Q and r0, r1 ∈ I such that q0 < t < r0 < r1 [u, r1] is(π, q0)-homogeneous (which exists by Lemma 0.19); and

I an automorphism α which is the identity outside (u, r1) andhas α(r0) < q0 (which exists by the homogeneity of Q).

More shifting and splitting, 1/3

TheoremIf G is sufficiently (?) generic, then L(G ) is a thicket.

For w ∈ I ∪ N and π a condition, say w is π-huge if for each〈q, n〉 ∈ a and each 〈x , y , k〉 ∈ B we have w > q, n, x , y , k .Fix sufficiently (?) generic G and π = 〈a,B〉 ∈ G withq0, c ∈ dom(a) such that

π ΦGe : 〈L(G); c〉 ≥s [q0, q1]G.

Have the following:

I an s ∈ I with s > max(dom(a));

I a positive natural N > max(ran(a) ∪ ran(B));

I a t ∈ Q and r0, r1 ∈ I such that q0 < t < r0 < r1 [u, r1] is(π, q0)-homogeneous (which exists by Lemma 0.19); and

I an automorphism α which is the identity outside (u, r1) andhas α(r0) < q0 (which exists by the homogeneity of Q).

More shifting and splitting, 1/3

TheoremIf G is sufficiently (?) generic, then L(G ) is a thicket.

For w ∈ I ∪ N and π a condition, say w is π-huge if for each〈q, n〉 ∈ a and each 〈x , y , k〉 ∈ B we have w > q, n, x , y , k .Fix sufficiently (?) generic G and π = 〈a,B〉 ∈ G withq0, c ∈ dom(a) such that

π ΦGe : 〈L(G); c〉 ≥s [q0, q1]G.

Have the following:

I an s ∈ I with s > max(dom(a));

I a positive natural N > max(ran(a) ∪ ran(B));

I a t ∈ Q and r0, r1 ∈ I such that q0 < t < r0 < r1 [u, r1] is(π, q0)-homogeneous (which exists by Lemma 0.19); and

I an automorphism α which is the identity outside (u, r1) andhas α(r0) < q0 (which exists by the homogeneity of Q).

More shifting and splitting, 2/3

π forces an unwanted computation, and we’ve chosen some usefulauxiliary objects.Define two extensions of π as follows:

π+ = 〈a ∪ {〈t,N〉, 〈α−1(q0), a(q0)〉},B ∪ {〈r0, s,N〉}〉,

π− = α(π+) = 〈a∪{〈α(t),N〉, 〈α(q0), a(q0)〉},B∪{〈α(r0), s,N〉}〉.

These are indeed conditions via useful choices above (in particular,note that α(s) = s).Let H = α(G ). Have π ∈ H and WLOG π+ ∈ G so π− ∈ H.

More shifting and splitting, 2/3

π forces an unwanted computation, and we’ve chosen some usefulauxiliary objects.Define two extensions of π as follows:

π+ = 〈a ∪ {〈t,N〉, 〈α−1(q0), a(q0)〉},B ∪ {〈r0, s,N〉}〉,

π− = α(π+) = 〈a∪{〈α(t),N〉, 〈α(q0), a(q0)〉},B∪{〈α(r0), s,N〉}〉.

These are indeed conditions via useful choices above (in particular,note that α(s) = s).Let H = α(G ). Have π ∈ H and WLOG π+ ∈ G so π− ∈ H.

More shifting and splitting, 2/3

π forces an unwanted computation, and we’ve chosen some usefulauxiliary objects.Define two extensions of π as follows:

π+ = 〈a ∪ {〈t,N〉, 〈α−1(q0), a(q0)〉},B ∪ {〈r0, s,N〉}〉,

π− = α(π+) = 〈a∪{〈α(t),N〉, 〈α(q0), a(q0)〉},B∪{〈α(r0), s,N〉}〉.

These are indeed conditions via useful choices above (in particular,note that α(s) = s).Let H = α(G ). Have π ∈ H and WLOG π+ ∈ G so π− ∈ H.

More shifting and splitting, 2/3

π forces an unwanted computation, and we’ve chosen some usefulauxiliary objects.Define two extensions of π as follows:

π+ = 〈a ∪ {〈t,N〉, 〈α−1(q0), a(q0)〉},B ∪ {〈r0, s,N〉}〉,

π− = α(π+) = 〈a∪{〈α(t),N〉, 〈α(q0), a(q0)〉},B∪{〈α(r0), s,N〉}〉.

These are indeed conditions via useful choices above (in particular,note that α(s) = s).

Let H = α(G ). Have π ∈ H and WLOG π+ ∈ G so π− ∈ H.

More shifting and splitting, 2/3

π forces an unwanted computation, and we’ve chosen some usefulauxiliary objects.Define two extensions of π as follows:

π+ = 〈a ∪ {〈t,N〉, 〈α−1(q0), a(q0)〉},B ∪ {〈r0, s,N〉}〉,

π− = α(π+) = 〈a∪{〈α(t),N〉, 〈α(q0), a(q0)〉},B∪{〈α(r0), s,N〉}〉.

These are indeed conditions via useful choices above (in particular,note that α(s) = s).Let H = α(G ). Have π ∈ H and WLOG π+ ∈ G so π− ∈ H.

More shifting and splitting, 3/3

Comparing G and H, we have [q0, q1]H 6∼= [q0, q1]G (the latter hasa block of size N while the former doesn’t).Since π ∈ G ∩ H we get

Φe : 〈L(G ); c〉 ≥s [q0, q1]G

andΦe : 〈L(H); c〉 ≥s [q0, q1]H .

Since α(c) = c have 〈L(G ); c〉 ∼= 〈L(H); c〉. And this contradictsisomorphism-invariance of Φe .

More shifting and splitting, 3/3

Comparing G and H, we have [q0, q1]H 6∼= [q0, q1]G (the latter hasa block of size N while the former doesn’t).

Since π ∈ G ∩ H we get

Φe : 〈L(G ); c〉 ≥s [q0, q1]G

andΦe : 〈L(H); c〉 ≥s [q0, q1]H .

Since α(c) = c have 〈L(G ); c〉 ∼= 〈L(H); c〉. And this contradictsisomorphism-invariance of Φe .

More shifting and splitting, 3/3

Comparing G and H, we have [q0, q1]H 6∼= [q0, q1]G (the latter hasa block of size N while the former doesn’t).Since π ∈ G ∩ H we get

Φe : 〈L(G ); c〉 ≥s [q0, q1]G

andΦe : 〈L(H); c〉 ≥s [q0, q1]H .

Since α(c) = c have 〈L(G ); c〉 ∼= 〈L(H); c〉. And this contradictsisomorphism-invariance of Φe .

More shifting and splitting, 3/3

Comparing G and H, we have [q0, q1]H 6∼= [q0, q1]G (the latter hasa block of size N while the former doesn’t).Since π ∈ G ∩ H we get

Φe : 〈L(G ); c〉 ≥s [q0, q1]G

andΦe : 〈L(H); c〉 ≥s [q0, q1]H .

Since α(c) = c have 〈L(G ); c〉 ∼= 〈L(H); c〉. And this contradictsisomorphism-invariance of Φe .

How much genericity do we need?

Above we glossed over “sufficient genericity.”

Key point: in neither proof did we need to force isomorphism, justequality of invariants (“which bits are coded in?”) after simplemodifications (shifts and automorphisms). That’s low-levelarithmetic.

QuestionWhat are the Turing degrees of thickets? (Are there ∆0

2 thickets?Are there PA-degrees not computing thickets?)

Also, this generalizes beyond ≤s : any “reasonably-definable”uniform reducibility notion has thickets (possibly after assuminglarge cardinals).

How much genericity do we need?

Above we glossed over “sufficient genericity.”Key point: in neither proof did we need to force isomorphism, justequality of invariants (“which bits are coded in?”) after simplemodifications (shifts and automorphisms). That’s low-levelarithmetic.

QuestionWhat are the Turing degrees of thickets? (Are there ∆0

2 thickets?Are there PA-degrees not computing thickets?)

Also, this generalizes beyond ≤s : any “reasonably-definable”uniform reducibility notion has thickets (possibly after assuminglarge cardinals).

How much genericity do we need?

Above we glossed over “sufficient genericity.”Key point: in neither proof did we need to force isomorphism, justequality of invariants (“which bits are coded in?”) after simplemodifications (shifts and automorphisms). That’s low-levelarithmetic.

QuestionWhat are the Turing degrees of thickets? (Are there ∆0

2 thickets?Are there PA-degrees not computing thickets?)

Also, this generalizes beyond ≤s : any “reasonably-definable”uniform reducibility notion has thickets (possibly after assuminglarge cardinals).

How much genericity do we need?

Above we glossed over “sufficient genericity.”Key point: in neither proof did we need to force isomorphism, justequality of invariants (“which bits are coded in?”) after simplemodifications (shifts and automorphisms). That’s low-levelarithmetic.

QuestionWhat are the Turing degrees of thickets? (Are there ∆0

2 thickets?Are there PA-degrees not computing thickets?)

Also, this generalizes beyond ≤s : any “reasonably-definable”uniform reducibility notion has thickets (possibly after assuminglarge cardinals).

Back to ordinals

So we can build several kinds of wild linear orders.But the ordinal situation is still unsatisfactory, and the crucial roleof automorphisms above means that all the preceding work isirrelevant to that.

A concrete problem to help things along:

ProblemEither find a pair of Medvedev-incomparable ordinals below thefirst recursively inaccessible, or show that no such exists.

Back to ordinals

So we can build several kinds of wild linear orders.But the ordinal situation is still unsatisfactory, and the crucial roleof automorphisms above means that all the preceding work isirrelevant to that.A concrete problem to help things along:

ProblemEither find a pair of Medvedev-incomparable ordinals below thefirst recursively inaccessible, or show that no such exists.