If 𝒞 is a pseudo-variety, then a supersimple residually 𝒞 group is nilpotent-by-poly-𝒞.

Si 𝒞 est une pseudo-variété, alors un groupe supersimple résiduellement 𝒞 est nilpotent-par-poly-𝒞.

Results in recursion-theoretic inductive inference have been criticized as depending on unrealistic self-referential examples. J. M. Bārzdiņš proposed a way of ruling out such examples, and conjectured that one of the earliest results of inductive inference theory would fall if his method were used. In this paper we refute Bārzdiņš' conjecture.

We propose a new line of research examining robust separations; these are defined using a strengthening of Bārzdiņš' original idea. The preliminary results of the new line of research are presented, and the most important open problem is stated as a conjecture. Finally, we discuss the extension of this work from function learning to formal language learning.

We prove that certain pairs of ordered structures are dependent. Among these structures are dense and tame pairs of o-minimal structures and further the real field with a multiplicative subgroup with the Mann property, regardless of whether it is dense or discrete.

We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of first-order topological theories.

Let κ be a regular uncountable cardinal and λ be a cardinal greater than κ. We show that if 2^<κ≤ M(κ,λ), then ◊_κ,λ holds, where M(κ,λ) equals λ^ℵ_₀ if cf(λ)≥ κ, and (λ⁺)^ℵ_₀ otherwise.

We locate winning strategies for various Σ⁰₃-games in the L-hierarchy in order to prove the following:

Theorem 1. KP + Σ₂-Comprehension ⊢ ∃ α L_α ⊨“Σ₂-KP + Σ⁰₃-Determinacy.”

Alternatively: Π¹₃-CA₀ ⊢“there is a β-model of Δ¹₃-CA₀ + Σ⁰₃-Determinacy.” The implication is not reversible. (The antecedent here may be replaced with Π¹₃(Π¹₃)-CA₀: Π¹₃ instances of Comprehension with only Π¹₃-lightface definable parameters—or even weaker theories.)

Theorem 2. KP + Δ₂-Comprehension + Σ₂-Replacement + AQI ⊬ Σ⁰₃-Determinacy.

(Here AQI is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively:

Δ¹₃CA₀ + AQI ⊬ Σ⁰₃-Determinacy.

Hence the theories: Π¹₃-CA₀, Δ¹₃-CA₀+ Σ⁰₃-Det, Δ¹₃-CA₀+AQI, and Δ¹₃-CA₀ are in strictly descending order of strength.

We prove that the set of all Polish groups admitting a compatible complete left-invariant metric (called CLI) is coanalytic non-Borel as a subset of a standard Borel space of all Polish groups. As an application of this result, we show that there does not exist a weakly universal CLI group. This, in particular, answers in the negative a question of H.Becker.

We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has non-empty interior, and any theory of pure tree is dp-minimal.

We show that there is a complete, consistent Borel theory which has no “Borel model” in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel.

We also investigate Borel isomorphisms between Borel structures.

Assuming the existence of a weakly compact hypermeasurable cardinal we prove that in some forcing extension ℵ_ω is a strong limit cardinal and ℵ_ω+2 has the tree property. This improves a result of Matthew Foreman (see [2]).

We study inversions of the jump operator on Π⁰₁ classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2-random sets which are not 2-random, and the jumps of the weakly 1-random relative to 0' sets which are not 2-random. Both of the classes coincide with the degrees above 0' which are not 0'-dominated. A further application is the complete solution of [24, Problem 3.6.9]: one direction of van Lambalgen's theorem holds for weak 2-randomness, while the other fails.

Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [24, Problem 8.2.14]: not all weakly 2-random sets are array computable. In fact, given any oracle X, there is a weakly 2-random which is not array computable relative to X. This contrasts with the fact that all 2-random sets are array computable.

One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V=L.

This paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α≤ω₁, that they are downward absolute to L for α <ω₁^L, and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.

We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.

Jullien's indecomposability theorem (INDEC) states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis, and then, in the base system RCA₀ plus Σ¹₁ induction, it was shown by Neeman to have strength strictly between weak Σ¹₁ choice and Δ¹₁ comprehension. We prove in this paper that Σ¹₁ induction is needed for the reversal of INDEC, that is for the proof that INDEC implies weak Σ¹₁ choice. This is in contrast with the typical situation in reverse mathematics, where reversals can usually be refined to use only Σ⁰₁ induction.

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) “If 𝒳 is a well-ordering, then so is ε_𝒳”, and (2) “If 𝒳 is a well-ordering, then so is φ(α,𝒳)”, where α is a fixed computable ordinal and φ represents the two-placed Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ACA₀⁺ over RCA₀. To prove the latter statement we need to use ω^α iterations of the Turing jump, and we show that the statement is equivalent to Π⁰_ω^α-CA₀. Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement “if 𝒳 is a well-ordering, then so is φ(𝒳,0)” is equivalent to ATR₀ over RCA₀.

We investigate the extension of monadic second-order logic of order with cardinality quantifiers “there exists uncountably many sets such that …” and “there exists continuum many sets such that …”. We prove that over the class of countable linear orders the two quantifiers are equivalent and can be effectively and uniformly eliminated. Weaker or partial elimination results are obtained for certain wider classes of chains. In particular, we show that over the class of ordinals the uncountability quantifier can be effectively and uniformly eliminated. Our argument makes use of Shelah's composition method and Ramsey-like theorem for dense linear orders.

The Infinite Time Turing Machine model [8] of Hamkins and Kidder is, in an essential sense, a “Σ₂-machine” in that it uses a Σ₂ Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate Σ_n rule. Such machines either halt or enter an infinite loop by stage ζ(n) =_df μ ζ(n) [∃ Σ(n) > ζ(n) L_ζ(n) ≺_Σn L_Σ(n)], again generalising precisely the ITTM case.

The collection of such machines taken together computes precisely those reals of the least model of analysis.

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. Recently, the author showed the consistency of ERNA with several transfer principles and proved results of nonstandard analysis in the resulting theories (see [12] and [13]). Here, we show that Weak König's lemma (WKL) and many of its equivalent formulations over RCA₀ from Reverse Mathematics (see [21] and [22]) can be ‘pushed down' into the weak theory ERNA, while preserving the equivalences, but at the price of replacing equality with equality ‘up to infinitesimals'. It turns out that ERNA plays the role of RCA₀ and that transfer for universal formulas corresponds to WKL.

We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let 𝒫 be P with its “induced ∅-definable structure”. The conditions are that 𝒫 (or rather its theory) is “rosy”, P has NIP in T and that P is stably 1-embedded in T. This generalizes a recent result of Hasson and Onshuus [6] which deals with the case where P is o-minimal in T. Our proofs make use of the theory of strict nonforking and weight in NIP theories ([3], [10]).

We build on an existing a term-sequent logic for the λ-calculus. We formulate a general sequent system that fully integrates αβη-reductions between untyped λ-terms into first order logic.

We prove a cut-elimination result and then offer an application of cut-elimination by giving a notion of uniform proof for λ-terms. We suggest how this allows us to view the calculus of untyped αβ-reductions as a logic programming language (as well as a functional programming language, as it is traditionally seen).

We prove that there is a Δ₂⁰, 1-random set Y such that every computably enumerable set which is computable from Y is strongly jump-traceable.

We also show that for every order function h there is an ω-c.e. random set Y such that every computably enumerable set which is computable from Y is h-jump-traceable. This establishes a correspondence between rates of jump-traceability and computability from ω-c.e. random sets.

We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example:

Theorem A. Det(Π¹₁-IND) → there exists an inner model with a strong cardinal.

Theorem B. Det(AQI) → there exist type-1 mice and hence inner models with proper classes of strong cardinals.

where Π¹₁-IND(AQI) is the pointclass of boldface Π¹₁-inductive (respectively arithmetically quasi-inductive) sets of reals.

We prove the following version of Hechler's classical theorem: For each partially ordered set (Q,≤) with the property that every countable subset of Q has a strict upper bound in Q, there is a ccc forcing notion such that in the generic extension for each tall analytic P-ideal ℐ (coded in the ground model) a cofinal subset of (ℐ,⊆^*) is order isomorphic to (Q,≤).