Shepherdson [14] showed that for a discrete ordered ring I, I is a model of IOpen iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ₄), then R must be recursively saturated. In particular, the real closure of I, RC(I), is recursively saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA.

In this paper, we continue the construction done in [3], so that under model-4-CA or 4-CA, given a bounded quadrangle C induced from a group configuration, we build a canonical hyperdefinable homogeneous space equivalent to C. When C is principal, we can choose the homogeneous space principal as well.

We show that if G is a strongly minimal finitely axiomatizable group, the division ring of quasi-endomorphisms of G must be an infinite finitely presented ring.

There is a proper countable support iteration of length ω adding no new reals at finite stages and adding a Sacks real in the limit.

This paper presents a polarized phase semantics, with respect to which the linear fragment of second order polarized linear logic of Laurent [15] is complete. This is done by adding a topological structure to Girard's phase semantics [9]. The topological structure results naturally from the categorical construction developed by Hamano—Scott [12]. The polarity shifting operator ↓ (resp. ↑) is interpreted as an interior (resp. closure) operator in such a manner that positive (resp. negative) formulas correspond to open (resp. closed) facts. By accommodating the exponentials of linear logic, our model is extended to the polarized fragment of the second order linear logic. Strong forms of completeness theorems are given to yield cut-eliminations for the both second order systems. As an application of our semantics, the first order conservativity of linear logic is studied over its polarized fragment of Laurent [16]. Using a counter model construction, the extension of this conservativity is shown to fail into the second order, whose solution is posed as an open problem in [16]. After this negative result, a second order conservativity theorem is proved for an eta expanded fragment of the second order linear logic, which fragment retains a focalized sequent property of [3].

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, 𝒟_h. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable, locally finite lattice) is isomorphic to an initial segment of 𝒟_h. Corollaries include the decidability of the two quantifier theory of 𝒟_h and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of ω₁^CK. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve ω₁. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of 𝒟_h.

We extend the notion of limitwise monotonic functions to include arbitrary computable domains. We then study which sets and degrees are support increasing (support strictly increasing) limitwise monotonic on various computable domains. As applications, we provide a characterization of the sets S with computable increasing η-representations using support increasing limitwise monotonic sets on ℚ and note relationships between the class of order-computable sets and the class of support increasing (support strictly increasing) limitwise monotonic sets on certain domains.

This paper aims to give a correct proof of Tait's conservative extension theorem. Tait's own proof is flawed in the sense that there are some invalid steps in his argument, and there is a counterexample to the main theorem from which the conservative extension theorem is supposed to follow. However, an analysis of Tait's basic idea suggests a correct proof of the conservative extension theorem and a corrected version of the main theorem.

Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result?

The primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊨φ (if and) only if Σ⊢φ∸ 2^-n for all n < ω. This approximated form of strong completeness asserts that if Σ⊨φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ.

Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theory T is decidable if for every sentence φ, the value φ_T is a recursive real, and moreover, uniformly computable from φ. If T is incomplete, we say it is decidable if for every sentence φ the real number φ_T^∘ is uniformly recursive from φ, where φ_T^∘is the maximal value of φ consistent with T. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable.

We study the connection between Schnorr triviality and genericity. We show that while no 2-generic is Turing equivalent to a Schnorr trivial and no 1-generic is tt-equivalent to a Schnorr trivial, there is a 1-generic that is Turing equivalent to a Schnorr trivial. However, every such 1-generic must be high. As a corollary, we prove that not all K-trivials are Schnorr trivial. We also use these techniques to extend a previous result and show that the bases of cones of Schnorr trivial Turing degrees are precisely those whose jumps are at least 0''.

Let ℳ=〈 M, +, <, 0, S〉 be a linear o-minimal expansion of an ordered group, and G=〈 G, ⊕, e_G〉 an n-dimensional group definable in ℳ. We show that if G is definably connected with respect to the t-topology, then it is definably isomorphic to a definable quotient group U/L, for some convex ⋁-definable subgroup U of 〈 Mⁿ, +〉 and a lattice L of rank equal to the dimension of the ‘compact part' of G.

Dp-minimality is a common generalization of weak minimality and weak o-minimality. If T is a weakly o-minimal theory then it is dp-minimal (Fact 2.2), but there are dp-minimal densely ordered groups that are not weakly o-minimal. We introduce the even more general notion of inp-minimality and prove that in an inp-minimal densely ordered group, every definable unary function is a union of finitely many continuous locally monotonic functions (Theorem 3.2).

The undecidability of first-order logic implies that there is no computable bound on the length of shortest proofs of valid sentences of first-order logic. Some valid sentences can only have quite long proofs. How hard is it to prove such “hard” valid sentences? The polynomial time tractability of this problem would imply the fixed-parameter tractability of the parameterized problem that, given a natural number n in unary as input and a first-order sentence φ as parameter, asks whether φ has a proof of length ≤ n. As the underlying classical problem has been considered by Gödel we denote this problem by p-Gödel. We show that p-Gödel is not fixed-parameter tractable if DTIME(h^O(1)) ≠ NTIME(h^O(1)) for all time constructible and increasing functions h. Moreover we analyze the complexity of the construction problem associated with p-Gödel.

We work in set-theory without choice ZF. A set is countable if it is finite or equipotent with ℕ. Given a closed subset F of [0,1]^I which is a bounded subset of l¹(I) (resp. such that F ⊆ c₀(I)), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice ZF_ℕ) implies that F is compact. This enhances previous results where ZF_ℕ (resp. the axiom of Dependent Choices) was required. If I is linearly orderable (for example I=ℝ), then, in ZF, the closed unit ball of the Hilbert space l²(I) is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of l²(𝒫(ℝ)) is not provable in ZF.

There exists a model in a countable language having a unique element which is not definable in ℒ_{ω₁, ω}.

Let ℛ be an o-minimal structure over a real closed field R. Given a simplicial complex K and some definable subsets S₁,..., S_l of its realization |K| in R we prove that there exist a subdivision K' of K and a definable triangulation φ':|K'|→ |K| of |K| partitioning S₁,...,S_l with φ' definably homotopic to id_|K|. As an application of this result we obtain the semialgebraic Hauptvermutung.

By taking a closer look at the construction of an Ackermann function we see that between any primitive recursive degree and its Ackermann modification there is a dense chain of primitive recursive degrees.

We obtain an equiconsistency for a weak form of universal indestructibility for strongness. The equiconsistency is relative to a cardinal weaker in consistency strength than a Woodin cardinal, Stewart Baldwin's notion of hyperstrong cardinal. We also briefly indicate how our methods are applicable to universal indestructibility for supercompactness and strong compactness.

In [9] we have considered a wide class of “well-behaved” reducibilities for sets of reals. In this paper we continue with the study of Borel reducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [9] allowing to deal with a larger class of notions of reduction (including, among others, the Baire class ξ functions).

In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the “finitary” infinite pigeonhole principle FIPP₁. That turned out to not be the proper formulation and so we proposed an alternative version FIPP₂. Tao himself formulated yet another version FIPP₃ in a revised version of his essay.

We give a counterexample to FIPP₁ and discuss for both of the versions FIPP₂ and FIPP₃ the faithfulness of their respective finitization of IPP by studying the equivalences IPP ↔ FIPP₂ and IPP ↔ FIPP₃ in the context of reverse mathematics ([9]). In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao's notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e., in terms of the “big five” subsystems of second order arithmetic.

Let G be a group definable in a monster model 𝔠 of a rosy theory satisfying NIP. Assume that G has hereditarily finitely satisfiable generics and 1 < U^þ(G) < ∞. We prove that if G acts definably on a definable set of U^þ-rank 1, then, under some general assumption about this action, there is an infinite field interpretable in 𝔠. We conclude that if G is not solvable-by-finite and it acts faithfully and definably on a definable set of U^þ-rank 1, then there is an infinite field interpretable in 𝔠. As an immediate consequence, we get that if G has a definable subgroup H such that U^þ(G)=U^þ(H)+1 and G/⋂_{g ∈ G}H^g is not solvable-by-finite, then an infinite field interpretable in 𝔠 also exists.

We prove a number of results in effective randomness, using methods in which Π⁰₁ classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.