The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V (in the sense that it correctly computes successors of singular cardinals greater than δ) or HOD is “far” from V (in the sense that all regular cardinals greater than or equal to δ are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that (assuming the existence of an extendible cardinal) that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails.

In 1952, Heinrich Scholz published a question in The Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser in turn asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached by means of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally by means of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open.

We survey the theory of Mučnik (weak) and Medvedev (strong) degrees of subsets of ωω with particular attention to the degrees of subsets of ω2. Sections 1–6 present the major definitions and results in a uniform notation. Sections 7–16 present proofs, some more complete than others, of the major results of the subject together with much of the required background material.

This article investigates the weak distributivity of Boolean σ-algebras satisfying the countable chain condition. It addresses primarily the question when such algebras carry a σ-additive measure. We use as a starting point the problem of John von Neumann stated in 1937 in the Scottish Book. He asked if the countable chain condition and weak distributivity are sufficient for the existence of such a measure.

Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent results provide a complete solution of both the set theoretic and the combinatorial problems. We shall survey the history of von Neumann's Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam to whom we dedicate this article.

§1. Complete Boolean algebras and weak distributivity. A Boolean algebra is a set B with Boolean operations a ˅ b (join), a ˄ b (meet) and −a (complement), partial ordering a ≤ b defined by a ˄ b = a and the smallest and greatest element, 0 and 1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty set S, under operations a ∪ b, a ∩ b, S − a, ordered by inclusion, with 0 = ∅ and 1 = S.

Complete Boolean algebras and weak distributivity.A Boolean algebra is a set B with Boolean operations a ˅ b (join), a ˄ b (meet) and -a (complement), partial ordering a ≤ b defined by a ˄ b = a and the smallest and greatest element. 0 and 1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty set S, under operations a ∪ b, a ∩ b, S -a, ordered by inclusion, with 0 = ϕ and 1 = S.

This paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main systems of reverse mathematics. While retaining the usual base theory and working still within second order arithmetic, theorems are described that range from those far below the usual systems to ones far above.

In 1936 Tarski sketched a rigorous definition of the concept of logical consequence which, he claimed, agreed quite well with common usage—or, as he also said, with the common concept of consequence. Commentators of Tarski's paper have usually been elusive as to what this common concept is. However, being clear on this issue is important to decide whether Tarski's definition failed (as Etchemendy has contended) or succeeded (as most commentators maintain). I argue that the common concept of consequence that Tarski tried to characterize is not some general, all-purpose notion of consequence, but a rather precise one, namely the concept of consequence at play in axiomatics. I identify this concept and show that Tarski's definition is fully adequate to it.

We represent truth sets for a variety of the well known semantic theories of truth as those sets consisting of all sentences for which a player has a winning strategy in an infinite two person game. The classifications of the games considered here are simple, those over the natural model of arithmetic being all within the arithmetical class of .

We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of Felix Klein's Erlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and the Martin Invariance Conjecture on their Turing degrees, i.e., their information content with respect to relative computability (Turing reducibility).

The philosopher Rudolf Carnap (1891–1970), although not himself an originator of mathematical advances in logic, was much involved in the development of the subject. He was the most important and deepest philosopher of the Vienna Circle of logical positivists, or, to use the label Carnap later preferred, logical empiricists. It was Carnap who gave the most fully developed and sophisticated form to the linguistic doctrine of logical and mathematical truth: the view that the truths of mathematics and logic do not describe some Platonistic realm, but rather are artifacts of the way we establish a language in which to speak of the factual, empirical world, fallouts of the representational capacity of language. (This view has its roots in Wittgenstein's Tractatus, but Wittgenstein's remarks on mathematics beyond first-order logic are notoriously sparse and cryptic.) Carnap was also the thinker who, after Russell, most emphasized the importance of modern logic, and the distinctive advances it enables in the foundations of mathematics, to contemporary philosophy. It was through Carnap's urgings, abetted by Hans Hahn, once Carnap arrived in Vienna as Privatdozent in philosophy in 1926, that the Vienna Circle began to take logic seriously and that positivist philosophy began to grapple with the question of how an account of mathematics compatible with empiricism can be given (see Goldfarb 1996).

A particular facet of Carnap's influence is not widely appreciated: it was Carnap who introduced Kurt Gödel to logic, in the serious sense. Although Gödel seems to have attended a course of Schlick's on philosophy of mathematics in 1925–26, his second year at the University, he did not at that time pursue logic further, nor did the seminar leave much of a trace on him. In the early summer of 1928, however, Carnap gave two lectures to the Circle which Gödel attended, or so I surmise. At these occasions, Carnap presented material from his manuscript treatise, Untersuchungen zur allgemeinen Axiomatik, that is, “Investigations into general axiomatics”, which dealt with questions of consistency, completeness and categoricity. Carnap later circulated this material to various people including Gödel.

In 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of relation algebras. He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and Jónsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarski's question is negative. Monk proved later that the answer remains negative even if one adjoins finitely many new axioms to Tarski's system. In this paper we describe a far-reaching generalization of the positive results of Jónsson and Tarski, as well as of some later, related results of Maddux. We construct a class of concrete models of Tarski's axioms—called coset relation algebras—that are very close in spirit to algebras of binary relations, but are built using systems of groups and cosets instead of elements of a base set. The models include all algebras of binary relations, and many non-representable relation algebras as well. We prove that every atomic relation algebra satisfying a certain measurability condition—a condition generalizing the conditions imposed by Jónsson and Tarski—is essentially isomorphic to a coset relation algebra. The theorem raises the possibility of providing a positive solution to Tarski's problem by using coset relation algebras instead of the standard algebras of binary relations.

We point out how the “Fundamental Theorem of Stability Theory”, namely the equivalence between the “non order property” and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck’s “Critères de compacité” from 1952. The familiar forms for the defining formulae then follow using Mazur’s Lemma regarding weak convergence in Banach spaces.

In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing.

The construction of a systematic philosophical foundation for logic is a notoriously difficult problem. In Part One I suggest that the problem is in large part methodological, having to do with the common philosophical conception of “providing a foundation”. I offer an alternative to the common methodology which combines a strong foundational requirement (veridical justification) with the use of non-traditional, holistic tools to achieve this result. In Part Two I delineate an outline of a foundation for logic, employing the new methodology. The outline is based on an investigation of why logic requires a veridical justification, i.e., a justification which involves the world and not just the mind, and what features or aspect of the world logic is grounded in. Logic, the investigation suggests, is grounded in the formal aspect of reality, and the outline proposes an account of this aspect, the way it both constrains and enables logic (gives rise to logical truths and consequences), logic's role in our overall system of knowledge, the relation between logic and mathematics, the normativity of logic, the characteristic traits of logic, and error and revision in logic.

We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the constants are not “closed” in suitable senses. We also improve the definitions and results on generalized strongly normal extensions from [Pillay, “Differential Galois theory I”, Illinois Journal of Mathematics, 42(4), 1998], using this to give a restatement of a conjecture on almost semiabelian δ-groups from [Bertrand and Pillay, “Galois theory, functional Lindemann–Weierstrass, and Manin maps”, Pacific Journal of Mathematics, 281(1), 2016].

The well known Wiener-Kuratowski explicit definition of the ordered pair, which sets (x,y) = {{x}, {x,y}}, works well in many set theories but fails for those with classes which cannot be members of singletons. With the aid of the Axiom of Foundation, we propose a recursive definition of ordered pair which addresses this shortcoming and also naturally generalizes to ordered tuples of greater length. There are many advantages to the new definition, for it allows for uniform definitions working equally well in a wide range of models for set theories. In ZFC and closely related theories, the rank of an ordered pair of two infinite sets under the new definition turns out to be equal to the maximum of the ranks of the sets.

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz”(“Relativism in Set Theory and the So-Called Theorem of Skolem”) in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world of objects that could only be grasped adequately by infinitary means. So the refutation might serve as a final clue to his epistemological credo.

Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic (cf. [8]). However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, (cf. [1]) he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”.

We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus these applications are not necessarily direct or expected.

The Löwenheim–Skolem theorem was published in Skolem's long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal notation is given here by which these proofs become transparent. A third section of Skolem's paper gives an analysis for derivations in plane projective geometry. To clear a gap in Skolem's result, a new conservativity property is shown for projective geometry, to the effect that a proper use of the axiom that gives the uniqueness of connecting lines and intersection points requires a conclusion with proper cases (logically, a disjunction in a positive part) to be proved.

The forgotten parts of Skolem's first paper on the Löwenheim–Skolem theorem are the perhaps earliest combinatorial analyses of formal mathematical proofs, and at least the earliest analyses with profound results.

Every K-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem.

Tarski's 1936 paper, “On the concept of logical consequence”, is a rather philosophical, non-technical paper that leaves room for conflicting interpretations. My purpose is to review some important issues that explicitly or implicitly constitute its themes. My discussion contains four sections: (1) terminological and conceptual preliminaries, (2) Tarski's definition of the concept of logical consequence, (3) Tarski's discussion of omega-incomplete theories, and (4) concluding remarks concerning the kind of conception that Tarski's definition was intended to explicate. The third section involves subsidiary issues, such as Tarski's discussion concerning the distinction between material and formal consequence and the important question ofthe criterion for distinguishing between logical and non-logical terms.

§1. Preliminaries. In this paper an argument is a two-part system composed of a set of propositions P (the premise-set) and a single proposition c (the conclusion). The expression ‘c is a [logical] consequence of P’ is used with the same meaning as the expression ‘c is [logically] implied by P’. The expressions ‘is a logical consequence of’ and the converse ‘implies’ are relational. Often, I shall be talking in the same sense of validity of an argument. Validity is a property of arguments; an argument with premise-set P and conclusion c is valid if and only if P implies c; i.e., c is a logical consequence of P. Notice that this notion of argument is strictly ontic; it does not involve any agent that thinks, determines or establishes that a given proposition is or is not a consequence of a given set of propositions.