Computable Structure Theory of Uncountable Linear Orders

Computable Structure Theoryof

Uncountable Linear Orders

Asher M. Kach

(Joint Work with Noam Greenberg, Steffen Lempp, and Daniel Turetsky)

Cornell University4 September 2012

Asher M. Kach Uncountable Linear Orders 4 September 2012 1 / 42

Outline

1 The Setting: Uncountable Computability Theory

2 Background Order Types: Uncountable Linear Orders

3 Computable Categoricity

4 Transfer Theorems

5 Degree Spectra

6 External RelationsThe Successor RelationThe Infinitely Far Apart Relation

7 Open Questions: The Uncountable Setting


Church’s Thesis in the ω1-Setting...

RemarkFor simplifying reasons, we assume V = L.

As a consequence, we have R ⊂ L(ω1) and ℵ1 = 2ℵ0 .

Theorem (Kripke; Platek; Sacks)The following are equivalent for a set A ⊆ ω1:

The set A is decidable by a Turing Machine with a tape oflength ω1 and for which halting computations are allowed to run forcountably many steps.Membership in A can be deduced in countably many steps frombasic axioms in a reasonable deduction calculus.The set A is ∆0

1-definable over L(ω1) with parameters.


Church’s Thesis in the ω1-Setting...

RemarkFor simplifying reasons, we assume V = L.

As a consequence, we have R ⊂ L(ω1) and ℵ1 = 2ℵ0 .

Theorem (Kripke; Platek; Sacks)The following are equivalent for a set A ⊆ ω1:

The set A is decidable by a Turing Machine with a tape oflength ω1 and for which halting computations are allowed to run forcountably many steps.Membership in A can be deduced in countably many steps frombasic axioms in a reasonable deduction calculus.The set A is ∆0

1-definable over L(ω1) with parameters.


Computably Enumerable Sets...

DefinitionA set S ⊂ ω is computably enumerable if it is Σ0

1.

A set S ⊂ ω is computable if both S and S are computablyenumerable. A (partial) function f is computable if its graph iscomputably enumerable. A structure S is computable if the relationsand functions on it are uniformly computable.

DefinitionA set S ⊆ Lα is α-computably enumerable if it is Σ0

1(Lα).

A set S ⊆ Lα is α-computable if both S and S are α-computablyenumerable. A (partial) function f is α-computable if its graph isα-computably enumerable. A structure S is α-computable if therelations and functions on it are uniformly α-computable.


Computably Enumerable Sets...

DefinitionA set S ⊂ ω is computably enumerable if it is Σ0

1.

A set S ⊂ ω is computable if both S and S are computablyenumerable. A (partial) function f is computable if its graph iscomputably enumerable. A structure S is computable if the relationsand functions on it are uniformly computable.

DefinitionA set S ⊆ Lα is α-computably enumerable if it is Σ0

1(Lα).

A set S ⊆ Lα is α-computable if both S and S are α-computablyenumerable. A (partial) function f is α-computable if its graph isα-computably enumerable. A structure S is α-computable if therelations and functions on it are uniformly α-computable.


The Uncountable Setting...

RemarkWhen developing computability theory with larger ordinals, attention isnormally restricted to admissible limit ordinals. Recall every successorcardinal is an admissible ordinal.

ConventionHenceforth, we (almost exclusively) restrict attention to when α is ω1.Thus, the term uncountable refers to objects of size ℵ1 (not larger).

RemarkTwo ideas are worth emphasizing:

Computations halt in countably many stages or run foruncountably many stages.Finite in ω-computability is similar to countable in ω1-computability;infinite in ω-computability is similar to uncountable inω1-computability.














Outline




4 Transfer Theorems

5 Degree Spectra




Dense Linear Orders

DefinitionDenote by η (also η0) the (unique) countable order type for which, forall sets A,B ⊂ η with |A|, |B| < ℵ0 and A < B, there are points x , y ,and z with x < A < y < B < z.

RemarkNote the familiar definition requires |A| = 1 and |B| = 1. These areequivalent as maximums and minimums of finite sets always exist.

DefinitionDenote by η1 the (unique) uncountable order type for which, for all setsA,B ⊂ η1 with |A|, |B| < ℵ1 and A < B, there are points x , y , and z withx < A < y < B < z.


Dense Linear Orders

DefinitionDenote by η (also η0) the (unique) countable order type for which, forall sets A,B ⊂ η with |A|, |B| < ℵ0 and A < B, there are points x , y ,and z with x < A < y < B < z.

RemarkNote the familiar definition requires |A| = 1 and |B| = 1. These areequivalent as maximums and minimums of finite sets always exist.

DefinitionDenote by η1 the (unique) uncountable order type for which, for all setsA,B ⊂ η1 with |A|, |B| < ℵ1 and A < B, there are points x , y , and z withx < A < y < B < z.


Dense Linear Orders

DefinitionDenote by ρ the order type of the real numbers, i.e., the completion ofthe order type η.

RemarkWe view ω ⊂ η ⊂ ρ as an artifact of N ⊂ Q ⊂ R.

Note ρ 6∼= η1. For example, take A = {0} and B = {q ∈ η : q > 0}.Alternately, take B = {q ∈ η}.

RemarkAs a consequence of the V = L assumption, we have |ρ| = ℵ1.


Dense Linear Orders






Dense Linear Orders






The Dushnik-Miller Theorem...

Theorem (Dushnik and Miller)Let L be a countably infinite linear order. Then L has a nontrivialself-embedding, i.e., there is an injection f : L → L that is not bijective.

Theorem (Dushnik and Miller)There is an uncountable linear order of size ℵ1 with no nontrivialself-embeddings.


The Dushnik-Miller Theorem...

Theorem (Dushnik and Miller)Let L be a countably infinite linear order. Then L has a nontrivialself-embedding, i.e., there is an injection f : L → L that is not bijective.

Theorem (Dushnik and Miller)There is an uncountable linear order of size ℵ1 with no nontrivialself-embeddings.


Outline




4 Transfer Theorems

5 Degree Spectra




Terminology...

DefinitionA computable structure S is computably categorical if there is acomputable isomorphism between any two computablepresentations A and B of S.

DefinitionA computable structure S is weakly uniformly computably categorical ifthere is a partial computable function f (i , j) such that Φf (i,j) is anisomorphism betweenMi andMj , whenever i and j are indices forcomputable presentationsMi andMj of S.


Terminology...

DefinitionA computable structure S is computably categorical if there is acomputable isomorphism between any two computablepresentations A and B of S.

DefinitionA computable structure S is weakly uniformly computably categorical ifthere is a partial computable function f (i , j) such that Φf (i,j) is anisomorphism betweenMi andMj , whenever i and j are indices forcomputable presentationsMi andMj of S.


The Countable Setting: A Characterization...

Theorem (Dzgoev; Remmel)A computable linear order is computably categorical if and only if it hasfinitely many adjacencies.

CorollaryA computable linear order is weakly uniformly computably categorical ifand only if it is finite or has order type η0.



Theorem (Dzgoev; Remmel)A computable linear order is computably categorical if and only if it hasfinitely many adjacencies.

CorollaryA computable linear order is weakly uniformly computably categorical ifand only if it is finite or has order type η0.


The Uncountable Setting: A Conjecture Refuted...

ConjectureA computable linear order is weakly uniformly computably categorical ifand only if it is countable or has order type η1.

ExampleThe order type η0 is not weakly uniformly computably categorical.

Indeed, as a consequence of the Dushnik-Miller Theorem, nocountably infinite linear order is weakly uniformly computablycategorical.

Theorem (Greenberg, Kach, Lempp, and Turetsky)A computable linear order is weakly uniformly computably categorical ifand only if it is finite or has order type η1.





















ConjectureA computable linear order is computably categorical if and only if it hascountably many adjacencies.

Example (6=⇒)The linear order 2 · ρ is computably categorical.

Proof.Fix presentations A and B of the order type 2 · ρ. Fix the (doubled)rationals in both presentations as a countable set of parameters.Before defining the isomorphism on any point in an irrational interval,wait for both points to appear. Map both points to the correspondingirrational interval (preserving order).




Example (6=⇒)The linear order 2 · ρ is computably categorical.

Proof.Fix presentations A and B of the order type 2 · ρ. Fix the (doubled)rationals in both presentations as a countable set of parameters.Before defining the isomorphism on any point in an irrational interval,wait for both points to appear. Map both points to the correspondingirrational interval (preserving order).




Example (6⇐=)The linear order η ∪ η · (ρ\η) is not computably categorical.

Proof.Construct a nonstandard copy A not computably isomorphic to astandard copy B. Defeat a candidate isomorphism ϕe : A → B bywaiting for it to be defined and a bijection from the eth copy of η in A tosome copy of η in B. Add a point to the eth interval of A.


The Uncountable Setting: A Characterization...

Theorem (Greenberg, Kach, Lempp, and Turetsky)A computable linear order is computably categorical if and only if thereis a countable parameter p and (uniformly) computable disjoint sets{Sn}n>0 such that every p-interval is finite or has order type η1 andevery p-interval of size n > 0 is in Sn.

Proof (⇐=).Fix such a parameter p and sets {Sn}n>0. Fix computable copies Aand B of L. Identify pA and pB.

Define π(pA) = pB. For arbitrary x ∈ A, identify the pA-interval towhich x belongs. Wait for the pA-interval to become infinite or beenumerated into some Sn. If the latter, wait for n many points. Mapvia π in an order-preserving manner. If more points appear, or if theformer, back-and-forth to build an isomorphism between copiesof η1.



Theorem (Greenberg, Kach, Lempp, and Turetsky)A computable linear order is computably categorical if and only if thereis a countable parameter p and (uniformly) computable disjoint sets{Sn}n>0 such that every p-interval is finite or has order type η1 andevery p-interval of size n > 0 is in Sn.

Proof (⇐=).Fix such a parameter p and sets {Sn}n>0. Fix computable copies Aand B of L. Identify pA and pB.

Define π(pA) = pB. For arbitrary x ∈ A, identify the pA-interval towhich x belongs. Wait for the pA-interval to become infinite or beenumerated into some Sn. If the latter, wait for n many points. Mapvia π in an order-preserving manner. If more points appear, or if theformer, back-and-forth to build an isomorphism between copiesof η1.



Proof (=⇒).Fixing a computable presentation A of a computably categorical linearorder, attempt to build B isomorphic but not computably isomorphic.

Strategy Φe: Inherits a countable parameter p.

1 Divide into p-intervals.





2 Wait for a p-interval, say I, to become nonempty.





3 Wait for Φe to converge on I. Enumerate I into Sn if I is finite andS1 otherwise.





4 Force I infinite or kill Φe.





4 Force I infinite or kill Φe:1 Wait for another point to appear in the interval.





4 Force I infinite or kill Φe:2 Add this point to attempt to defeat the isomorphism.





4 Force I infinite or kill Φe:3 Continue to add points to attempt to defeat the isomorphism.





5 Force infinite I nonscattered or kill Φe.





5 Force infinite I nonscattered or kill Φe:1 Locate the oldest infinite block.





5 Force infinite I nonscattered or kill Φe:2 Wait for Φe to converge on this block.





5 Force infinite I nonscattered or kill Φe:3 Double this block.





5 Force infinite I nonscattered or kill Φe:4 Wait for the opposing interval to catch up. Repeat ω many

times.Asher M. Kach Uncountable Linear Orders 4 September 2012 16 / 42




6 Force nonscattered I to be η1 or kill Φe.





6 Force nonscattered I to be η1 or kill Φe:1 Cycle through all pairs of countable sets.





6 Force nonscattered I to be η1 or kill Φe:2 Place a point between the sets, while maintaining isomorphism.





6 Force nonscattered I to be η1 or kill Φe:3 Wait for the opposing interval to catch up. Repeat for the next pair

of countable sets.Asher M. Kach Uncountable Linear Orders 4 September 2012 16 / 42

Theorems from The Book...?

Theorem (Dzgoev; Remmel)A computable linear order is computably categorical if and only if thereis a finite parameter p and (uniformly) (computable) disjoint sets{Sn}n>0 such that every p interval is finite or has order type η0 andevery p interval of size n > 0 is in Sn.

Theorem (Greenberg, Kach, Lempp, and Turetsky)A computable linear order is computably categorical if and only if thereis a countable parameter p and (uniformly) computable disjoint sets{Sn}n>0 such that every p interval is finite or has order type η1 andevery p interval of size n > 0 is in Sn.

Corollary (Dushnik and Miller)There is an ω2-computably categorical linear order having an ℵ1-sizedparameter p such that no p-interval is finite or has order type η2.


Theorems from The Book...?

Theorem (Dzgoev; Remmel)A computable linear order is computably categorical if and only if thereis a finite parameter p and (uniformly) (computable) disjoint sets{Sn}n>0 such that every p interval is finite or has order type η0 andevery p interval of size n > 0 is in Sn.

Theorem (Greenberg, Kach, Lempp, and Turetsky)A computable linear order is computably categorical if and only if thereis a countable parameter p and (uniformly) computable disjoint sets{Sn}n>0 such that every p interval is finite or has order type η1 andevery p interval of size n > 0 is in Sn.

Corollary (Dushnik and Miller)There is an ω2-computably categorical linear order having an ℵ1-sizedparameter p such that no p-interval is finite or has order type η2.


Outline




4 Transfer Theorems

5 Degree Spectra




Transfer Theorems in the Countable Setting...

Theorem (Downey and Knight)If λ is 0′-computable, then (η0 + 2 + η0) · λ is computable.

Theorem (Watnick; Ash)If λ is 0′′-computable, then ω · λ is computable.

If λ is 0(2α)-computable, then ωα · λ is computable.


Transfer Theorems in the Uncountable Setting...

Theorem (Greenberg, Kach, Lempp, and Turetsky)For any degree a > 0, there is an a-computable order type λ such thatκ · λ is not computable for any (nonempty) order-type κ.

DefinitionFix X ⊆ ω1. Define λX to be the order type of η + X ⊆ ρ, having fixingan effective identification between ω1 and ρ\η.

Lemma (Greenberg and Knight)

The order type λX is d-computable if and only if X ∈ Σ01(d).

Proof of Theorem.Fix A ∈ a. Take λ := λA⊕A. Show, for nonempty κ, thatDegSpec(κ · λ) = DegSpec(κ) ∩ DegSpec(λ).
















A Pseudo-Transfer Theorem...

DefinitionDefine Lc to be the linear ordering formed from L by filling with onepoint every cut such that the set of points to the left of the cut has atmost countable cofinality.

Define Lt to be the linear ordering

Lt :=∑x∈Lc

Zx ,

where Zx := 2 if x ∈ L and Zx := η1 if x ∈ Lc − L.

Theorem (Greenberg, Kach, Lempp, and Turetsky)

Fix a degree a. A linear ordering L is a′-computable if and only if Lt isa-computable.


A Pseudo-Transfer Theorem...

Proof (⇐=).Given an a-computable presentation of Lt , let K := Succ(Lt ). Then Kis a′-computable and has order type L under the induced order.

Proof (=⇒).View L as being a (a′-computable) subset of a fixed computable copyof η1. Approximate this subset. Build Lt , using the intervals of ordertype η1 for garbage collection.


Outline




4 Transfer Theorems

5 Degree Spectra




Terminology...

DefinitionA linear order L has degree a if the set

DegSpec(L) := {d : L has a d-computable copy}

has least degree a.

DefinitionFix a computable ordinal α. A linear order L has αth jump degree a ifthe set

{d(α) : L has a d-computable copy}

has least degree a.


Terminology...

DefinitionA linear order L has degree a if the set

DegSpec(L) := {d : L has a d-computable copy}

has least degree a.

DefinitionFix a computable ordinal α. A linear order L has αth jump degree a ifthe set

{d(α) : L has a d-computable copy}

has least degree a.


Degree Spectra Examples...

Theorem (R. Miller)There is a noncomputable linear order whose degree spectrumcontains all hyperimmune degrees.

Theorem (Greenberg, Kach, Lempp, and Turetsky)There is a noncomputable linear order whose degree spectrumcontains all hyperimmune degrees.

Theorem (Greenberg, Kach, Lempp, and Turetsky)For every positive integer n, there is an order-type whose degreespectrum consists of exactly the non-lown degrees.













Theorem (Various: Ash, Jockusch, Knight, Richter, and Soare)If a linear order has degree, it has degree 0. There is a linear orderwith degree.

If a linear order has first jump degree, it has first jump degree 0′.There is a linear order with proper first jump degree.

For every computable ordinal α ≥ 2 and degree a ≥ 0(α), there is alinear order with proper αth jump degree a.



ConjectureIf a linear order has degree, it has degree 0.

If a linear order has first jump degree, it has first jump degree 0′.


Example (Greenberg and Knight)Recall that, for any set A, the linear order LX is Z -computable if andonly if X is Σ0

1(Z ).

Fixing A ∈ a, it follows the linear order LA⊕A has degree a.



ConjectureIf a linear order has degree, it has degree 0.

If a linear order has first jump degree, it has first jump degree 0′.


Example (Greenberg and Knight)Recall that, for any set A, the linear order LX is Z -computable if andonly if X is Σ0

1(Z ).

Fixing A ∈ a, it follows the linear order LA⊕A has degree a.



Theorem (Greenberg; Greenberg, Kach, Lempp, and Turetsky)

For every integer n and degree a ≥ 0(n), there is a linear order withproper nth jump degree a.

Proof.For n = 0: The linear order LA⊕A suffices.

For n = 1: Fix b with a = b′. Relativize the construction of thehyperimmune linear order to b, yielding K. Take K + 1 + LB⊕B.

For n ≥ 2: Exploit (relativize) the existence of an order type whosedegree spectra is precisely the nonlow degrees. Invoke the TransferTheorem.


Outline




4 Transfer Theorems

5 Degree Spectra




The Successor Relation...

DefinitionIf L is a linear order, points x , y ∈ L are successors if the intervalstrictly between x and y is empty.

If x and y are successors, we write Succ(x , y); if A is a presentation ofa linear order L, we write Succ(A) for the set {(x , y) : Succ(x , y)}.

If L is a computable linear order, we denote the set of degrees

{deg(Succ(A)) : A is a computable presentation of L}

by DegSpecSucc(L).

RemarkNote that DegSpecSucc(L) consists only of computably enumerabledegrees since Succ(x , y) is Π0

1.


The Successor Relation...

DefinitionIf L is a linear order, points x , y ∈ L are successors if the intervalstrictly between x and y is empty.

If x and y are successors, we write Succ(x , y); if A is a presentation ofa linear order L, we write Succ(A) for the set {(x , y) : Succ(x , y)}.


{deg(Succ(A)) : A is a computable presentation of L}

by DegSpecSucc(L).

RemarkNote that DegSpecSucc(L) consists only of computably enumerabledegrees since Succ(x , y) is Π0

1.


The Countable Setting...

Theorem (Downey, Lempp, and Wu)For every linear order L with infinitely many successivities, the setDegSpecSucc(L) is upward closed within the computably enumerabledegrees.

Theorem (Downey and Moses)For every computably enumerable degree a, there is a linear order Lwith a the least degree in DegSpecSucc(L).



Theorem (Downey, Lempp, and Wu)For every linear order L with infinitely many successivities, the setDegSpecSucc(L) is upward closed within the computably enumerabledegrees.

Theorem (Downey and Moses)For every computably enumerable degree a, there is a linear order Lwith a the least degree in DegSpecSucc(L).



ConjectureFor every linear order L with uncountably many successivities, the setDegSpecSucc(L) is upward closed within the computably enumerabledegrees.

ExampleThe linear order 2 · ρ has DegSpecSucc(L) = {0}.

Proposition (Greenberg, Kach, Lempp, and Turetsky)For any computably enumerable degree a, there is a computable linearorder L with DegSpecSucc(L) = {a}.

Proof.Fix A ∈ a computably enumerable. The linear order L replacing the x thirrational by 2 if x 6∈ A and by η if x ∈ A suffices.














The Uncountable Setting: The Theorem...

Proposition (Greenberg, Kach, Lempp, and Turetsky)There is a computable linear order L withDegSpecSucc(L) = {a : a is computably enumerable}.

Proof.The linear order L replacing the 〈x , y〉th irrational by 2 if x 6∈Wy andby 3 if x ∈Wy suffices.

Theorem (Greenberg, Kach, Lempp, and Turetsky)If L is a computable linear order with the property that for anycountable parameter p, there is a p-interval which is neither finite nordense, then the set DegSpecSucc(L) is upwards closed within thecomputably enumerable degrees.


The Uncountable Setting: The Theorem...

Proposition (Greenberg, Kach, Lempp, and Turetsky)There is a computable linear order L withDegSpecSucc(L) = {a : a is computably enumerable}.

Proof.The linear order L replacing the 〈x , y〉th irrational by 2 if x 6∈Wy andby 3 if x ∈Wy suffices.

Theorem (Greenberg, Kach, Lempp, and Turetsky)If L is a computable linear order with the property that for anycountable parameter p, there is a p-interval which is neither finite nordense, then the set DegSpecSucc(L) is upwards closed within thecomputably enumerable degrees.


The Uncountable Setting: Some Limits...

DefinitionFix a computable linear order having a countable parameter p forwhich every p-interval is finite or dense.

Define Ip>1(L) to be the set of p-intervals containing more than one

element.

Define Ipn (L) to be the set of p-intervals (as a subset of Ip

>1(L))containing precisely n elements.

Define Ip∞(L) to be the set of p-intervals (as a subset of Ip

>1(L))containing infinitely many elements.


The Uncountable Setting: Some Limits...

Theorem (Greenberg, Kach, Lempp, and Turetsky)If A is a computable presentation of a computable linear order Lhaving a countable parameter p for which every p-interval is finite ordense, then

Ip∞(A) ≤T Succ(B) ≤wtt

⊕n>1

Ipn (A).

for all computable presentations B of L.

Theorem (Greenberg, Kach, Lempp, and Turetsky)There exists a linear order L in which not every degree consistent withthis is realized.


The Infinitely Far Apart Relation...

DefinitionIf L is a linear order, points x , y ∈ L are infinitely far apart if the intervalstrictly between x and y is infinite.

If x and y are infinitely far apart, we write Inf(x , y); if A is a presentationof a linear order L, we write Inf(A) for the set {(x , y) : Inf(x , y)}.


{deg(Inf(A)) : A is a computable presentation of L}

by DegSpecInf(L).



RemarkNote that DegSpecInf(L) consists of Π0

2 degrees since Inf(x , y) is Π02.

RemarkInteresting results exist.



RemarkNote that DegSpecInf(L) consists only of computably enumerabledegrees since Inf(x , y) is Σ0

1.

Proposition (Greenberg, Kach, Lempp, and Turetsky)For every computably enumerable degree a, there is a computablelinear order L with DegSpecInf(L) = {a}.

Proposition (Greenberg, Kach, Lempp, and Turetsky)There is a computable linear order L such thatDegSpecInf(L) = {a : a is computably enumerable}.




1.






1.




Outline




4 Transfer Theorems

5 Degree Spectra




Questions...

QuestionIs there a noncomputable linear order L having presentations in allnoncomputable degrees?

QuestionAre there reasonable characterizations of the 0(n) computablycategorical linear orders?

QuestionWhat (more) can be said about DegSpecSucc(L), DegSpecInf(L),and DegSpecUnc(L)?

QuestionWhat can be said about the computable structure theory and effectivealgebra of other classes of algebraic structures?


Questions...






Questions...






Questions...






Questions...

QuestionThere are a number of differences in the computable structure theoryof linear orders between the ω setting and the ω1 setting.

Do these differences continue to manifest themselves at higher(admissible) cardinals?

QuestionFor countable, uncountable, and larger linear orders, what is thecorrect characterization of computable categoricity?


Questions...

QuestionThere are a number of differences in the computable structure theoryof linear orders between the ω setting and the ω1 setting.

Do these differences continue to manifest themselves at higher(admissible) cardinals?

QuestionFor countable, uncountable, and larger linear orders, what is thecorrect characterization of computable categoricity?


Questions...

QuestionAfter developing an appropriate definition for 0(α) for α ≥ ω, what canbe said about proper αth jump degrees?


ReferencesRod Downey, Steffen Lempp, and Guohua Wu.On the complexity of the successivity relation in computable linear orderings.J. Math. Log., 10(1-2):83–99, 2010.

Rodney Downey and Julia F. Knight.

Orderings with αth jump degree 0(α).Proc. Amer. Math. Soc., 114(2):545–552, 1992.

Rodney G. Downey and Michael F. Moses.Recursive linear orders with incomplete successivities.Trans. Amer. Math. Soc., 326(2):653–668, 1991.

Ben Dushnik and E. W. Miller.Concerning similarity transformations of linearly ordered sets.Bull. Amer. Math. Soc., 46:322–326, 1940.

Noam Greenberg, Asher M. Kach, Steffen Lempp, and Daniel D. Turetsky.Computability and uncountable linear orders.submitted.

Noam Greenberg and Julia F. Knight.Computable structure theory using admissible recursion theory on ω1.In Effective Mathematics of the Uncountable, Lecture Notes in Logic. Association for Symbolic Logic, Cambridge, to appear.

J. B. Remmel.Recursive isomorphism types of recursive Boolean algebras.J. Symbolic Logic, 46(3):572–594, 1981.

Joseph G. Rosenstein.Linear orderings, volume 98 of Pure and Applied Mathematics.Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.


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Computable Structure Theory of Uncountable Linear Orders

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