Prime Models of Computably Enumerable Degree
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1 Prime Models of Computably Enumerable Degree Department of Mathematics University of Chicago July 3, 2008 / Logic Colloquium, Bern
2 In 1961, Robert Vaught introduced prime, homogeneous, and saturated models. During the 1970 s, two groups separately began to study the computability of these models: Harrington, Millar, and Morley in America Goncharov, Nurtazin, and Peretyat kin in Russia For this talk, all languages, theories, and structures are countable, and all theories are complete.
3 In 1961, Robert Vaught introduced prime, homogeneous, and saturated models. During the 1970 s, two groups separately began to study the computability of these models: Harrington, Millar, and Morley in America Goncharov, Nurtazin, and Peretyat kin in Russia For this talk, all languages, theories, and structures are countable, and all theories are complete.
4 In 1961, Robert Vaught introduced prime, homogeneous, and saturated models. During the 1970 s, two groups separately began to study the computability of these models: Harrington, Millar, and Morley in America Goncharov, Nurtazin, and Peretyat kin in Russia For this talk, all languages, theories, and structures are countable, and all theories are complete.
5 Definition (Prime Model) A model A of a theory T is prime if A elementarily embeds into every model of T. For example, the algebraic numbers are a prime model of ACF 0. Definition (Complete Formula) An n-ary formula θ(x) consistent with T is complete if for all ψ(x), either T (θ ψ) or T (θ ψ). Definition (Atomic Model) A model A of T is atomic if each tuple in A satisfies a complete formula. Thus, an atomic model realizes only the principal types of T. A countable model A is atomic A is prime.
6 Definition (Prime Model) A model A of a theory T is prime if A elementarily embeds into every model of T. For example, the algebraic numbers are a prime model of ACF 0. Definition (Complete Formula) An n-ary formula θ(x) consistent with T is complete if for all ψ(x), either T (θ ψ) or T (θ ψ). Definition (Atomic Model) A model A of T is atomic if each tuple in A satisfies a complete formula. Thus, an atomic model realizes only the principal types of T. A countable model A is atomic A is prime.
7 Definition (Prime Model) A model A of a theory T is prime if A elementarily embeds into every model of T. For example, the algebraic numbers are a prime model of ACF 0. Definition (Complete Formula) An n-ary formula θ(x) consistent with T is complete if for all ψ(x), either T (θ ψ) or T (θ ψ). Definition (Atomic Model) A model A of T is atomic if each tuple in A satisfies a complete formula. Thus, an atomic model realizes only the principal types of T. A countable model A is atomic A is prime.
8 Definition (Prime Model) A model A of a theory T is prime if A elementarily embeds into every model of T. For example, the algebraic numbers are a prime model of ACF 0. Definition (Complete Formula) An n-ary formula θ(x) consistent with T is complete if for all ψ(x), either T (θ ψ) or T (θ ψ). Definition (Atomic Model) A model A of T is atomic if each tuple in A satisfies a complete formula. Thus, an atomic model realizes only the principal types of T. A countable model A is atomic A is prime.
9 Definition (Atomic Theory) A theory T is atomic if every formula consistent with T can be extended to a complete formula. Theorem (Prime Model Theorem, Vaught) Every complete atomic theory has a prime model. How can we effectivize this theorem?
10 Definition (Atomic Theory) A theory T is atomic if every formula consistent with T can be extended to a complete formula. Theorem (Prime Model Theorem, Vaught) Every complete atomic theory has a prime model. How can we effectivize this theorem?
11 Definition (Atomic Theory) A theory T is atomic if every formula consistent with T can be extended to a complete formula. Theorem (Prime Model Theorem, Vaught) Every complete atomic theory has a prime model. How can we effectivize this theorem?
12 Definition (Decidable Theories and Models) A theory T is decidable if the set of sentences in T is computable. A model A is decidable if its elementary diagram D e (A) is computable. A model A has degree d if D e (A) has Turing degree d. Another interesting line of research is to study the atomic diagrams.
13 Theorem (Prime Model Theorem, Vaught) Every complete atomic theory has a prime model. The Prime Model Theorem cannot be effectivized: Theorem (Goncharov-Nurtazin (1974), Millar (1978)) There is a complete atomic decidable (CAD) theory with no decidable prime model. Proof idea: If A is a prime model, then D e (A) can compute a listing of the principal types of T. Build a CAD theory T so that if ϕ e appears to list the principal types, it lists some non-principal type as well.
14 Theorem (Prime Model Theorem, Vaught) Every complete atomic theory has a prime model. The Prime Model Theorem cannot be effectivized: Theorem (Goncharov-Nurtazin (1974), Millar (1978)) There is a complete atomic decidable (CAD) theory with no decidable prime model. Proof idea: If A is a prime model, then D e (A) can compute a listing of the principal types of T. Build a CAD theory T so that if ϕ e appears to list the principal types, it lists some non-principal type as well.
15 Theorem (Prime Model Theorem, Vaught) Every complete atomic theory has a prime model. The Prime Model Theorem cannot be effectivized: Theorem (Goncharov-Nurtazin (1974), Millar (1978)) There is a complete atomic decidable (CAD) theory with no decidable prime model. Proof idea: If A is a prime model, then D e (A) can compute a listing of the principal types of T. Build a CAD theory T so that if ϕ e appears to list the principal types, it lists some non-principal type as well.
16 Question What are the degrees of prime models of CAD theories? Theorem (Knight, Upward Closure, (1986)) For a nontrivial theory T, the set of degrees of prime models of T is upward closed. Theorem (Denisov (1989), Drobotun (1978), Millar (1978)) Every CAD theory has a prime model computable in 0. 0 can carry out the usual construction of a prime model. Many people have worked on improving this.
17 Question What are the degrees of prime models of CAD theories? Theorem (Knight, Upward Closure, (1986)) For a nontrivial theory T, the set of degrees of prime models of T is upward closed. Theorem (Denisov (1989), Drobotun (1978), Millar (1978)) Every CAD theory has a prime model computable in 0. 0 can carry out the usual construction of a prime model. Many people have worked on improving this.
18 Question What are the degrees of prime models of CAD theories? Theorem (Knight, Upward Closure, (1986)) For a nontrivial theory T, the set of degrees of prime models of T is upward closed. Theorem (Denisov (1989), Drobotun (1978), Millar (1978)) Every CAD theory has a prime model computable in 0. 0 can carry out the usual construction of a prime model. Many people have worked on improving this.
19 Question What are the degrees of prime models of CAD theories? Theorem (Knight, Upward Closure, (1986)) For a nontrivial theory T, the set of degrees of prime models of T is upward closed. Theorem (Denisov (1989), Drobotun (1978), Millar (1978)) Every CAD theory has a prime model computable in 0. 0 can carry out the usual construction of a prime model. Many people have worked on improving this.
20 Question What are the degrees of prime models of CAD theories? Theorem (Knight, Upward Closure, (1986)) For a nontrivial theory T, the set of degrees of prime models of T is upward closed. Theorem (Denisov (1989), Drobotun (1978), Millar (1978)) Every CAD theory has a prime model computable in 0. 0 can carry out the usual construction of a prime model. Many people have worked on improving this.
21 Theorem (Csima, Prime Low Basis Theorem (2004)) Every CAD theory has a prime model A of low degree, i.e., D e (A) T 0.
22 Computably Enumerable Sets Definition A set C is computably enumerable (c.e.) if there is an effective listing of its elements. Computably enumerable sets and degrees have played a central role in computability theory since Post s Problem of There are many applications of the c.e. sets, including: Unsolvability of Hilbert s 10th problem Unsolvability of word problem for groups Differential geometry
23 Computably Enumerable Sets Definition A set C is computably enumerable (c.e.) if there is an effective listing of its elements. Computably enumerable sets and degrees have played a central role in computability theory since Post s Problem of There are many applications of the c.e. sets, including: Unsolvability of Hilbert s 10th problem Unsolvability of word problem for groups Differential geometry
24 Computably Enumerable Sets Definition A set C is computably enumerable (c.e.) if there is an effective listing of its elements. Computably enumerable sets and degrees have played a central role in computability theory since Post s Problem of There are many applications of the c.e. sets, including: Unsolvability of Hilbert s 10th problem Unsolvability of word problem for groups Differential geometry
25 Remark If D e (A) is a c.e. set, then it is computable. We will instead focus our attention on prime models of computably enumerable Turing degree. Definition A Turing degree c is computably enumerable (c.e.) if c contains a c.e. set.
26 Remark If D e (A) is a c.e. set, then it is computable. We will instead focus our attention on prime models of computably enumerable Turing degree. Definition A Turing degree c is computably enumerable (c.e.) if c contains a c.e. set.
27 Theorem (Csima, Hirschfeldt, Knight, Soare, Prime Bounding (2004)) The degree d 0 is the degree of a prime model of every CAD theory d > 0 (i.e, d is nonlow 2 ). Corollary All nonlow 2 c.e. degrees are the degrees of prime models for every CAD theory.
28 Theorem (Csima, Hirschfeldt, Knight, Soare, Prime Bounding (2004)) The degree d 0 is the degree of a prime model of every CAD theory d > 0 (i.e, d is nonlow 2 ). Corollary All nonlow 2 c.e. degrees are the degrees of prime models for every CAD theory.
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30 Prime Low C.E. Basis Theorem Theorem (Epstein) Every CAD theory has a prime model of low c.e. degree. The proof differs greatly from Csima s proof for low degrees: Csima used a forcing argument with a 0 -oracle construction. For the c.e. case, we use a priority argument with a computable construction.
31 Prime Low C.E. Basis Theorem Theorem (Epstein) Every CAD theory has a prime model of low c.e. degree. The proof differs greatly from Csima s proof for low degrees: Csima used a forcing argument with a 0 -oracle construction. For the c.e. case, we use a priority argument with a computable construction.
32 Density of Prime Models of C.E. Degree Theorem (Sacks, Density Theorem) Let d and c be c.e. degrees with d < c. Then there is a c.e. degree b between d and c. Theorem (Epstein) Let T be a CAD theory, c be nonlow 2 and c.e., and d < c be c.e. Then there is a prime model A of T of c.e. degree a with d < a < c and a = d. Uses infinite injury Below any nonlow 2 c.e. degree, there is a prime model of low c.e. degree. Can make a any degree c.e. in c and above d.
33 Density of Prime Models of C.E. Degree Theorem (Sacks, Density Theorem) Let d and c be c.e. degrees with d < c. Then there is a c.e. degree b between d and c. Theorem (Epstein) Let T be a CAD theory, c be nonlow 2 and c.e., and d < c be c.e. Then there is a prime model A of T of c.e. degree a with d < a < c and a = d. Uses infinite injury Below any nonlow 2 c.e. degree, there is a prime model of low c.e. degree. Can make a any degree c.e. in c and above d.
34
35 Pushing Down the Degree of a Prime Model Theorem (Epstein) Let c > 0 be the c.e. degree of a prime model of a CAD theory T. Then there is a prime model of T of low c.e. degree a < c. This theorem does not hold for non-c.e. degrees.
36 Pushing Down the Degree of a Prime Model Theorem (Epstein) Let c > 0 be the c.e. degree of a prime model of a CAD theory T. Then there is a prime model of T of low c.e. degree a < c. This theorem does not hold for non-c.e. degrees.
37 Corollary For any degree c, with 0 < c < 0, every CAD theory T has a prime model of low c.e. degree a such that c a (i.e., c a and a c).
38 Corollary Every CAD T has a minimal pair of prime models of low c.e. degree.
39 Question Can we combine the pushdown and density theorems? That is, given a prime model of c.e. (low 2 ) degree c, and given a c.e. degree d < c, is there a prime model of c.e. degree a between d and c? Can we guarantee that a = d?
40 For Further Reading B. F. Csima. Degree Spectra of Prime Models. Journal of Symbolic Logic, 69: , B. F. Csima, D. R. Hirschfeldt, J. F. Knight, and R. I. Soare. Bounding Prime Models. Journal of Symbolic Logic, 69: , R. Epstein. Journal of Symbolic Logic, to appear. Thank you for listening.
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