Reverse Mathematics of First-Order Theories With Finitely Many Models

Belanger, David

doi:10.1017/jsl.2014.32

Cited by 6 publications

(4 citation statements)

References 11 publications

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“…In fact, the following example shows that Effectively Prime ⇒ Effectively Atomic does not always hold. We thank David Belanger for this observation which is connected to his paper [1].…”

mentioning

confidence: 68%

Effective prime uniqueness

Cholak

McCoy

2017

Proc. Amer. Math. Soc.

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Assuming the obvious definitions below, we show that a decidable model that is effectively prime is also effectively atomic. This implies that two effectively prime (decidable) models are computably isomorphic. This is in contrast to the theorem that there are two atomic decidable models which are not computably isomorphic. We end with a section describing the implications of this result in reverse mathematics.Proof. Use a back-and-forth construction with complete formulas determining how to extend the partial isomorphism. Theorem 2. (Atomic ⇒ Prime) If A is a countable atomic model of T , then A is prime.Proof. Use the "forth" half of the back-and-forth argument. Theorem 3. (Prime ⇒ Atomic) IfA is a prime model of T , then A is countable and atomic.Proof. By the compact theorem T has a countable model. Therefore A is countable. Let a ∈ A , and consider its type. For any other model B of T , there is an elementary embedding of A into B, so that B also realizes this type. By the Omitting Types Theorem, the type of a includes a complete formula.

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“…In fact, the following example shows that Effectively Prime ⇒ Effectively Atomic does not always hold. We thank David Belanger for this observation which is connected to his paper [1].…”

mentioning

confidence: 68%

Effective prime uniqueness

Cholak

McCoy

2017

Proc. Amer. Math. Soc.

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“…We now argue in RCA 0 to show that (3) implies B⌃ 0 2 , and hence so does (2). Assume that every strongly 1-homogeneous model is homogeneous.…”

Section: Defining Homogeneitymentioning

confidence: 89%

“…morphisms) required to verify that the definition holds of M. Other natural examples for investigation include issues dealing with categoricity for theories (as is common in model theory, rather than for structures as is common in computable model theory), universality, and saturation. Some of these questions have recently been investigated reverse mathematically by Belanger in [2] and [3].…”

Section: Introductionmentioning

confidence: 99%

Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem

2017

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“…We will discuss genericity below, but note here that the principle that for each A, there is an infinite tree T such that every path through T is 1-generic relative to A has the same properties. These are not the first examples of these kinds of "monsters in the reverse mathematics zoo": Belanger [8,9] found several natural model-theoretic principles equivalent to ¬WKL ∨ ACA.…”

Section: Rmyc-mmentioning

confidence: 99%

Coarse computability, the density metric, Hausdorff distances between Turing degrees, perfect trees, and reverse mathematics

Hirschfeldt¹,

Jockusch²,

Schupp³

2021

Preprint

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For A ⊆ ω, the coarse similarity class of A, denoted by [A], is the set of all B ⊆ ω such that the symmetric difference of A and B has asymptotic density 0. There is a natural metric δ on the space S of coarse similarity classes defined by letting δ([A], [B]) be the upper density of the symmetric difference of A and B. We study the metric space of coarse similarity classes under this metric, and show in particular that between any two distinct points in this space there are continuum many geodesic paths. We also study subspaces of the form {[A] : A ∈ U } where U is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of U .We then define a distance between Turing degrees based on Hausdorff distance in the metric space (S, δ). We adapt a proof of Monin to show that the Hausdorff distances between Turing degrees that occur are exactly 0, 1/2, and 1, and study which of these values occur most frequently in the senses of Lebesgue measure and Baire category. We define a degree a to be attractive if the class of all degrees at distance 1/2 from a has measure 1, and dispersive otherwise. In particular, we study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition for embeddability.Motivated by a couple of issues arising in the above work, we also study the computability-theoretic and reverse-mathematical aspects of a Ramseytheoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually 1-random, as well as a perfect set whose elements are mutually 1-generic.Finally, we study the completeness of (S, δ) from the perspectives of computability theory and reverse mathematics.

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Reverse Mathematics of First-Order Theories With Finitely Many Models

Cited by 6 publications

References 11 publications

Effective prime uniqueness

Effective prime uniqueness

Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem

Coarse computability, the density metric, Hausdorff distances between Turing degrees, perfect trees, and reverse mathematics

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