In 1963, the first author introduced a course in set theory at the University of Illinois whose main objectives were to cover Godel's work on the con sistency of the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH), and Cohen's work on the independence of the AC and the GCH. Notes taken in 1963 by the second author were taught by him in 1966, revised extensively, and are presented here as an introduction to axiomatic set theory. Texts (...) in set theory frequently develop the subject rapidly moving from key result to key result and suppressing many details. Advocates of the fast development claim at least two advantages. First, key results are high lighted, and second, the student who wishes to master the subject is com pelled to develop the detail on his own. However, an instructor using a "fast development" text must devote much class time to assisting his students in their efforts to bridge gaps in the text. (shrink)

In order to be able to express all possible patterns of dependence and independence between variables, we have to replace the traditional first-order logic by independence-friendly (IF) logic. Our natural concept of truth for a quantificational sentence S says that all the Skolem functions for S exist. This conception of truth for a sufficiently rich IF first-order language can be expressed in the same language. In a first-order axiomatic set theory, one can apparently express this same concept in set-theoretical (...) terms, since the existence of functions can be expressed there. Because of Tarski's theorem, this is impossible. Hence there must exist set-theoretical statements, even provable ones, which are said to be true in first-order models of axiomatic set theory but whose Skolem functions do not all exist. Hence there are provable sentences in axiomatic set theory that are false in accordance with our ordinary conceptions of set-theoretical truth. Such counter-intuitive propositions have been known to exist, but they have been blamed on the peculiarities of very large sets. It is argued here that this explanation is not correct and that there are intuitively false theorems not involving very large sets. Hence the provability or unprovability of a set-theoretical statement, e.g. of the continuum hypothesis (CH) in axiomatic set theory is not necessarily relevant to the truth of CH. (shrink)

Introduction. The system of axioms for set theory to be exhibited in this paper is a modification of the axiom system due to von Neumann. In particular it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property (“definite Eigenschaft”), which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary, in the sense of being formalizable in the logical calculus (...) of first order, which contains no other bound variables than individual variables and no accessory rule of inference (as, for instance, a scheme of complete induction).The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and to utilize at the same time some of the set-theoretic concepts of the Schröder logic and of Principia mathematica which have become familiar to logicians. As will be seen, a considerable simplification results from this arrangement.The theory is not set up as a pure formalism, but rather in the usual manner of elementary axiom theory, where we have to deal with propositions which are understood to have a meaning, and where the reference to the domain of facts to be axiomatized is suggested by the names for the kinds of individuals and for the fundamental predicates.On the other hand, from the formulation of the axioms and the methods used in making inferences from them, it will be obvious that the theory can be formalized by means of the logical calculus of first order (“Prädikatenkalkul” or “engere Funktionenkalkül”) with the addition of the formalism of equality and the ι-symbol for “descriptions” (in the sense of Whitehead and Russell). (shrink)

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of (...) infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.Let us recall the main points of this procedure.For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined. (shrink)

As is well known, some paradoxes arise through inadequate analysis of the meanings of terms in a language, an adequate analysis showing that the paradoxes arise through a lack of separation of an object theory and a metatheory. Under such an adequate analysis in which parts of the metatheory are modelled in the object theory, the paradoxes give way to remarkable theorems establishing limitations of the object theory.Such a modelling is often accomplished by a Gödel numbering. Here we shall (...) use a somewhat different technique in axiomatic set theory, from which we shall reap a few results having the effect of comparing the strength of various axiom schema of comprehension for sets and classes. Similar results were obtained by A. Mostowski [7] using Gödel numbering. (shrink)

This work applies the competitive exclusion principle and the concept of potential competitors as simple axiomatic tools to generalized situations in ecology. These tools enable apparent competition and its dual counterpart to be explicitly evaluated in poorly understood ecological systems. Within this set-theory framework we explore theoretical symmetries and invariances, De Morgan’s laws, frozen evolutionary diversity and virtual processes. In particular, we find that the exclusion principle compromises the geometrical growth of the number of species. By theoretical extending this (...) principle, we can describe interspecific depredation in the dual case. This study also briefly considers the debated situation of intraspecific competition. The ecological consequences of our findings are discussed; particularly, the use of our framework to reinterpret coupled mathematical differential equations describing certain ecological processes. (shrink)

Another exceptionally fine text by Suppes. Designed for those who can follow a mathematical argument, but presupposes no special knowledge of mathematics or symbolic logic. The system developed is that of Zermelo-Fraenkel. Especially noteworthy is the discussion of the exact role played by the various axioms.--N. D. B., Jr.

What has been the historical relationship between set theory and logic? On the one hand, Zermelo and other mathematicians developed set theory as a Hilbert-style axiomatic system. On the other hand, set theory influenced logic by suggesting to Schröder, Löwenheim and others the use of infinitely long expressions. The questions of which logic was appropriate for set theory - first-order logic, second-order logic, or an infinitary logic - culminated in a vigorous exchange between Zermelo and Gödel around 1930.

The principle _determinatio est negatio_—that determination is achieved through negation—has philosophical roots extending back to Plato and Aristotle, and it later influenced early modern thinkers such as Francisco Suárez and Spinoza. This paper has two aims. The first demonstrates how the principle of negation functions as a tool for conceptual determination across various philosophical frameworks, and the second demonstrates that the principle plays a key role in the analysis and resolution of the Burali-Forti paradox within the context of the naïve (...) and axiomatic set theories. In the first section, the analysis focuses on the evolution of the principle from ancient philosophy to early modern metaphysics, examining Plato’s dialectics, Aristotle’s metaphysical distinctions, Suárez’s scholastic theories, and Spinoza’s monist metaphysics. The second section shifts to mathematics, where _determinatio est negatio_ plays a key role in resolving set-theoretical paradoxes, particularly the Burali-Forti paradox. By exploring Cantor’s solution and its reliance on the distinction between consistent and inconsistent sets, this study demonstrates how this principle is essential for avoiding self-referential inconsistencies. The contributions of Zermelo and von Neumann, who developed axiomatic frameworks to address these paradoxes, further elaborate the philosophical foundations of set theory. Ultimately, the study reveals a deep connection between metaphysical principles and the mathematical treatment of paradoxes, emphasizing the ongoing relevance of _determinatio est negatio_ in both domains. (shrink)

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of (...) itself? What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski. In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable. (shrink)

We discuss several axiomatizations of set theory in first order predicate calculus with epsilon and a constant symbol W, starting with the simple system K(W) which has a strong equivalence with ZF without Foundation. The other systems correspond to various extensions of ZF by certain large cardinal hypotheses. These axiomatizations are unusually simple and uncluttered, and are highly suggestive of underlying philosophical principles that generate higher set theory.

In the first part of this paper, we consider several natural axioms in urelement set theory, including the Collection Principle, the Reflection Principle, the Dependent Choice scheme and its generalizations, as well as other axioms specifically concerning urelements. We prove that these axioms form a hierarchy over $\text {ZFCU}_{\text {R}}$ (ZFC with urelements formulated with Replacement) in terms of direct implication. The second part of the paper studies forcing over countable transitive models of $\text {ZFU}_{\text {R}}$. We propose a new (...) definition of ${\mathbb P}$ -names to address an issue with the existing approach. We then prove the fundamental theorem of forcing with urelements regarding axiom preservation. Moreover, we show that forcing can destroy and recover certain axioms within the previously established hierarchy. Finally, we demonstrate how ground model definability may fail when the ground model contains a proper class of urelements. (shrink)