Abstract: It is argued that set theory provides a powerful addition to commonsense reasoning, facilitating expression of meta-knowledge, names, and self-reference. Difficulties in establishing a suitable language to include sets for such purposes are discussed, as well as what appear to be promising solutions. Ackermann’s set theory as well as a more recent theory involving universal sets are discussed in terms of their relevance to commonsense.

Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a far-away planet. Would their mathematics be set-based? What are the alternatives to the set-theoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.

To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus ” for set theory, it is essentially based on just two set-theoretical principles: extensionality and comprehension (to which we add ∈-induction and optionally the axiom of choice). Comprehension is formulated as: x ∈{x | ϕ} ↔ϕ, where {x | ϕ} is a legal set term of the theory. In order for {x | ϕ} to be legal, ϕ should be safe with respect to {x}, where safety is a relation between formulas and finite sets of variables. The various systems we consider differ from each other mainly with respect to the safety relations they employ. These relations are all defined purely syntactically (using an induction on the logical structure of formulas). The basic one is based on the safety relation which implicitly underlies commercial query languages for relational database systems (like SQL). Our framework makes it possible to reduce all extensions by definitions to abbreviations. Hence it is very convenient for mechanical manipulations and for interactive theorem proving. It also provides a unified treatment of comprehension axioms and of absoluteness properties of formulas. 1

Set theory is sometimes formulated by starting with two sorts of entities called individuals and classes, and then defining a set to be a class as one, that is, a class which is at the same time an individual, as indicated

Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such principles justify the kind of cardinals needed for, inter alia, Woodin’s Ω-Logic. 1 To say that the universe of all sets is an unfinished totality does not mean objective undeterminateness, but merely a subjective inability to finish it. Gödel, in Wang, [17] 1 Reflection Principles in Set Theory Historically reflection principles are associated with attempts to say that no one notion, idea, or statement can capture our whole view of the universe of sets V = ⋃ α∈On Vα where On is the class of all ordinals. That no one idea can pin down the universe of all sets has firm historical roots (see the quotation from Cantor later or the following): The Universe of sets cannot be uniquely characterized (i.e. distinguished from all its initial segments) by any internal structural property of the membership relation in it, which is expressible in any logic of finite or transfinite type, including infinitary logics of any cardinal number. Gödel: Wang- ibid. Indeed once set theory was formalized by the (first order version of) the axioms and schemata of Zermelo with the additions of Skolem and Fraenkel, it was seen that reflection of first order formulae ϕ(v0, , vn) in the language of set theory L∈ ˙ could actually be proven:

The 20th century choice for the axioms 1 of Set Theory are the Zermelo-Frankel axioms together with the Axiom of Choice, these are the ZFC axioms. This particular choice has led to a 21th century problem: The ZFC Delemma: Many of the fundamental questions of Set Theory are formally unsolvable from the ZFC axioms. Perhaps the most famous example is given by the problem of the Continuum Hypothesis: Suppose X is an infinite set of real numbers, must it be the case that either X is countable or that the set X has cardinality equal to the cardinality of the set of all real numbers? One interpretation of this development is: Skeptic’s Attack: The Continuum Hypothesis is neither true nor false because the entire conception of the universe of sets is a complete fiction. Further, all the theorems of Set Theory are merely finitistic truths, a reflection of the mathematician and not of any genuine mathematical “reality”. Here and in what follows, the “Skeptic ” simply refers to the meta-mathematical position which denies any genuine meaning to a conception of uncountable sets. The counter-view is that of the “Set Theorist”: 1This paper is dedicated to the memory of Paul J. Cohen. 1 Gödel Book—input: Woodin (rev. 2009 Oct 04) 2

The 20th century witnessed the development and refinement of the mathematical notion of infinity. Here of course I am referring primarily to the development of Set Theory which is that area of modern mathematics devoted to the study of infinity. This development raises an obvious question: Is there a non-physical realm of infinity?

Abstract. Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a number