this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory.

Abstract. ‘One universe, one logic ’ takes the world as it is and leads to adjointness as the global logic of anything. The alternative approach to find a unification of known logics requires assumptions and is therefore consistent with the same conclusion for a universal logic has to be universally applicable. The universal characteristic of adjointness is that it has a natural construction from the concept of the arrow. The application to the test sentence, ‘John said that Mary believed he did not love her’, demonstrates adjointness as the logic of the post-modern world. 1 Unity of Applicable Logic There is one ultimate logic: it is a simple ontological but pragmatic argument of ‘one universe, one logic’. If more, how can we know unless there is a logic to compare them? If logic is a family of varying strength, what logic compares the variance? Only some ultimate logic. How do we even know this? It must still be the same logic that tells us this. And that logic must tell us about itself − − tell us that it has some recursive self-closure. The same pragmatic cogency leads us into the world of physics and beyond into the humanities. The world must fit together according to this same ultimate logic. It is therefore an applicable logic. Universal logic means universally applicable logic. This study arises from the investigation of fundamentals in two large applied areas: one is schema design in interoperable databases, the other is in legal reasoning; both studies relate logic to real-world facts. Until we are able to identify the ultimate logic of the universe, it is not surprising that goals like unified field theory within a ”theory of everything ” are so elusive. Applicable logic is needed in new ways in biology, medicine, economics, legal science, natural computing, modern physics, etc. This means it has to be a logic which can manage the advances made in the twentieth century, many of

Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Gödel in the intermediate generation and making contributions in proof theory, set theory, and the philosophy of mathematics. Bernays is best known for the two-volume 1934,1939 Grundlagen der Mathematik [39, 40], written solely by him though Hilbert was retained as first author. Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of first-order logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. 1 Recent re-evaluation of Bernays ’ role actually places him at the center of the development of mathematical logic and Hilbert’s program. 2 But starting in his forties, Bernays did his most individuated, distinctive mathematical work in set theory, providing a timely axiomatization and later applying higher-order reflection principles, and produced a stream of

The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measurable?”), cardinal arithmetic (“Does Cantor’s Continuum Hypothesis hold?”), combinatorics(“DoesSuslin’sHypotheseshold?”), andgrouptheory (“Is there a Whitehead group?”). These developments gave rise to two conflicting positions. The first position—which we shall call pluralism—maintains that the independence results largely undermine the enterprise of set theory as an objective enterprise. On this view, although there are practical reasons that one might give in favour of one set of axioms over another—say, that it is more useful for a given task—, there are no theoretical reasons that can be given; and, moreover, this either implies or is a consequence of the fact—depending on the variant of the view, in particular, whether it places realism before reason,

The independence results in arithmetic and set theory led to a proliferation of mathematical systems. One very general way to investigate the space of possible mathematical systems is under the relation of interpretability. Under this relation the space of possible mathematical systems forms an intricate hierarchy of increasingly strong systems. Large cardinal axioms provide a canonical means of climbing this hierarchy and they play a central role in comparing systems from conceptually distinct domains. This article is an introduction to independence, interpretability, large cardinals and their interrelations. Section 1 surveys the classic independence results in arithmetic and set theory. Section 2 introduces the interpretability hierarchy and describes some of its basic features. Section 3 introduces the notion of a large cardinal axiom and discusses some of the central examples. Section 4 brings together the previous themes by discussing the manner in which large cardinal axioms provide a canonical means for climbing the hierarchy of interpretability and serve as an intermediary in the comparison

The continuum hypotheses (CH) is one of the most central open problems in set theory, one that is important for both mathematical and philosophical reasons. The problem actually arose with the birth of set theory; indeed, in many respects it stimulated the birth of set theory. In 1874 Cantor had shown that there is a one-to-one correspondence between the natural numbers and the algebraic numbers. More surprisingly, he showed that there is no oneto-one correspondence between the natural numbers and the real numbers. Taking the existence of a one-to-one correspondence as a criterion for when two sets have the same size (something he certainly did by 1878), this result shows that there is more than one level of infinity and thus gave birth to the higher infinite in mathematics. Cantor immediately tried to determine whether there were any infinite sets of real numbers that were ofintermediate size, that is, whether there was an infinite set of real numbers that could not be put into one-to-one correspondence with the natural numbers and