William S. Hatcher

Zermelo–Fraenkel Set Theory

This chapter focuses on Zermelo-Fraenkel set theory (ZF). Zermelos answer to the paradoxes and inconsistencies in Freges system, as contained in his axioms for set theory, is based on the assumption that the sets which mathematicians need can be built up from certain simple sets by means of given operations. The approach is in many ways inverse to the approach of Poincare-Russell. Russells system is concerned with logical and linguistic principles such as impredicativity and the vicious-circle arguments; Zermelos system is more directly concerned with mathematics and the needs of mathematical structures. Relations of partial order are often represented by a two-dimensional display consisting of vertices, representing elements of the set S , and lines connecting the vertices with the understanding that if one vertex x is connected to another vertex y, y being situated above x , then x holds. Two vertices that are not connected are considered not comparable, regardless of their relative positions. If two vertices x and y are connected directly—without any intervening vertex, and if y , then this means that there is no element z such that y holds.

Hilbert's Program and Gödel's Incompleteness Theorems

This chapter focuses on Hilberts program and Godels incompleteness theorems. A consistency proof of any system is an absolute proof of consistency if the proof is based on no stronger assumptions than the assumption that S is consistent. Certain simpler theorems of logic, such as the consistency of any predicate calculus, can be used in an absolute proof of consistency. It was Hilbert who originally formulated the discipline of metamathematics, by which he meant the study of formal systems by weak, number-theoretic methods, and with the goal of giving an absolute consistency proof of a system in which all of mathematics can be deduced. Hilberts goal was to justify certain nonconstructive principles of mathematics, such as the axiom of choice or definition by transfinite induction. Intuitionism can be schematically—though in a somewhat oversimplified way—described as the position that only number-theoretic methods are valid mathematical tools. The rest, intuitionists would assert, is just a fiction.

The Origin of Modern Foundational Studies

This chapter discusses the origin of modern foundational studies. There are several fundamental points on which most mathematicians would agree regardless of their personal philosophic convictions concerning the nature of mathematics. The first is that mathematics is abstract, and that it consists primarily of reasoning with and contemplating abstractions. The second is that the truth or falsity of a proposition in mathematics is determined by a process of deduction, of showing that the proposition can be proved on the basis of some given first principles or assumed truths. This process seems to differ from other sciences in at least one respect—every other science, even one as abstract as physics, ultimately depends upon a certain amount of manipulation of the physical world. The intuitive set theory which generated the paradoxes had been initiated and developed by Cantor. His main contribution was the general theory of infinite sets, and Cantors notions were essential to the Dedekind-Weierstrass construction of the real numbers.

The Foundational Systems of W. V. Quine

Starting in 1937, W. V. Quine elaborated several systems capable of reproducing a reasonably large portion of mathematics. The point of departure for New Foundations (NF) is, to a great extent, the theory of types, particularly the system ST. Quine was strongly influenced by the work of Whitehead and Russell, and had long been seeking some way of liberalizing the type restriction to obtain a more satisfactory foundation. One theorem of logic deducible within any set theory such as Zermelo-Fraenkel set theory (ZF) is that a first-order theory is consistent, if and only if, it has a model. In particular, the axiom of choice can be disproved within the system, and Cantors paradox is narrowly avoided. Failure to deduce a contradiction in NF is no proof that the system is consistent. Fin is the set of all finite sets. A set is finite if it is an element of a natural number n regarded as a class of all n -element sets. Inf is the set of all infinite sets.

Chapter 8 – Categorical Algebra

Publisher Summary This chapter focuses on categorical algebra. Until rather recently, most mathematicians have carried on work in their special branch of mathematics without giving much thought to foundations. The most prevalent attitudes were that logicians had somewhere already solved the problem of giving a once-and-for-all foundation for mathematics or else that foundational problems were essentially irrelevant to mathematical practice anyway. The first idea has been undermined by the plethora of independence proofs in set theory starting with the work of Cohen. These results showed that there are many different, incompatible models of Zermelo-Fraenkel (ZF) set theory and thus, tended to destroy set theorys—largely unwarranted—image as an absolute foundation for mathematics. The second notion that foundational questions are irrelevant to mathematical practice, has also been considerably undermined. In particular, a number of problems in the theory of abelian groups depend on whether one assumes the existence of inaccessible cardinals in set theory.

First-order Logic

IN ORDER to understand clearly the formal languages to be presented in later chapters of this book, it will be necessary for the reader to have some knowledge of first-order logic. This chapter serves to furnish the necessary tools. The reader who is already familiar with these questions can easily treat this chapter as a review, though some attention should be given to our particular form of the rules for the predicate calculus, which will be used in the remainder of this study.

The Theory of Types

BEGINNING in 1908, Russell proposed a system called the theory of types, which was to serve as a foundation for mathematics. It represented Russells answer to the paradoxes and was founded on Russells vicious circle principle which we discussed in Chapter 3. The form of Russells theory presented in the famous Principia Mathematica written in collaboration with A. N. Whitehead is very complex and contains some partially conflicting tendencies which will be explained later. We shall develop first a theory PT of predicative type theory, weaker than the system PM of Principia Mathematica, but like it in form and spirit. We shall then proceed to type theory TT, a system essentially equivalent to the system PM but simpler in form. Finally, we shall present a simplified theory ST of simple types, which is a reformulation by Godel and Tarski of type theory. During the course of the discussion we shall consider briefly a system RT of ramified types.

Frege's System and the Paradoxes

BEGINNING in 1879, G. Frege [1] and [2] elaborated a formal system intended to serve as a foundation for mathematics. We shall present Freges system in a form quite close to the original except for extensive notational simplification.

Foundations as a Branch of Mathematics

It is quite clear the way in which we can, and perhaps should, consider mathematical logic as a branch of mathematics, in fact a part of applied mathematics. The main technique used in applied mathematics is that of studying a phenomenon by building a mathematical model of it. Such a model is an idealization which approximates the phenomenon by concentrating on a few relevant features and ignoring what seems to be less significant. If the model turns out to have high predictive and explanatory value, at least in some useful situations, it is subjected to extensive theoretical development and becomes a ‘theory’ (i.e. probability theory, heat theory, etc.). It is precisely in this way that mathematical logic can be viewed as applied mathematics. The phenomenon which underlies mathematical logic is common-sense logical inference. This phenomenon has several aspects, in particular, language, ‘reality’, meaning, and deduction. The mathematical model of this situation consists of the following abstractions : formal languages for language, mathematical structures for reality, interpretation in structures for meaning, and formal deduction for deduction. Though this model neglects many features of the actual phenomenon (most notably the psychology of the reasoner), it has been judged useful enough to be studied extensively. Now, if one accepts the above as a reasonable summary of what mathematical logic is, then mathematical logic is no more ‘philosophical than any other branch of applied mathematics. The only reason one might be tempted to argue otherwise is because in logic it is, in part, the human thought process itself which is being modelled, thus apparently rendering the model more ‘subjective’. Yet, if modern philosophical analysis has shown anything, it has shown just how much subjectivity enters into our models even of physical phenomena. Thus, logic would not seem to have any special status on this account. One could even argue that logic is more