Why Study Logic?

The term “logic” has always had two meanings – a broader one and a narrower one:

1. (1) All the general laws about the movement of human thinking should ultimately be made explicit so that thinking can be a reliable instrument, but

2. (2) already Aristotle realized that one must start on that vast program with a more sharply defined subcase.

The achievements of this subprogram include the recognition of the necessity of making explicit

1. (a) a limited universe of discourse, as well as

2. (b) the correspondence assigning, to each adjective that is meaningful over a whole universe, the part of that universe where the adjective applies. This correspondence necessarily involves

3. (c) an attendant homomorphic relation between connectives (like and and or) that apply to the adjectives and corresponding operations (like intersection and union) that apply to the parts “named” by the adjectives.

When thinking is temporarily limited to only one universe, the universe as such need not be mentioned; however, thinking actually involves relationships between several universes. For example, the three universes (1) of differential equations, (2) of functions of time, and (3) of formal power series are all distinct with different classes of adjectives meaningful over each one. But there are key relationships between these three universes that are the everyday preoccupation of users of many mathematical sciences: a function might satisfy a differential equation, a power series might approximate a function or might formally satisfy a differential equation, whereas we might seek to solve a differential equation or expand a function, and so on.

Why Sets for Mathematics?

This book is for students who are beginning the study of advanced mathematical subjects such as algebra, geometry, analysis, or combinatorics. A useful foundation for these subjects will be achieved by openly bringing out and studying what they have in common.

A significant part of what is common to all these subjects was made explicit 100 years ago by Richard Dedekind and Georg Cantor, and another significant part 50 years ago by Samuel Eilenberg and Saunders Mac Lane. The resulting idea of categories of sets is the main content of this book. It is worth the effort to study this idea because it provides a unified guide to approaching constructions and problems in the science of space and quantity.

More specifically, it has become standard practice to represent an object of mathematical interest (for example a surface in three-dimensional space) as a “structure.” This representation is possible by means of the following two steps:

1. (1) First we deplete the object of nearly all content. We could think of an idealized computer memory bank that has been erased, leaving only the pure locations (that could be filled with any new data that are relevant). The bag of pure points resulting from this process was called by Cantor a Kardinalzahl, but we will usually refer to it as an abstract set.

2. (2) Then, just as computers can be wired up in specific ways, suitable specific mappings between these structureless sets will constitute a structure that reflects the complicated content of a mathematical object. For example, the midpoint operation in Euclidean geometry is represented as a mapping whose “value” at any pair of points is a special third point.

The Axiom of Infinity: Number Theory

Recall that we denote by S the category of (abstract, discrete, constant) sets and arbitrary maps between them that we have studied till now. In the various branches of mathematics (such as mechanics, geometry, analysis, number theory, logic) there arise many different categories χ of (not necessarily discrete, variable) sets and respectful maps between them. The relation between S and the χ's is (at least) threefold:

1. (0) S is “case zero” of an χ in that in general the sets in χ have some sort of structure such as glue, motion and so on, but in S this structure is reduced to nothing. However, the general X often has a functor determining the mere number (Cantor) |X| of each such emergent aggregate X.

2. (1) A great many of the mathematical properties of such a category χ of variable sets are the same or similar to properties of the category S of constant sets. Thus, a thorough knowledge of the properties of S, together with some categorical wisdom, can be indispensable in dealing with problems of analysis, combinatorics, and so forth. The main common properties include the concepts of function spaces XT and of power sets P(X).

3. […]

The axiom of choice was first formulated by Zermelo in 1904 and used to prove his Well-Ordering theorem. The axiom was considered controversial because it introduced a highly nonconstructive aspect that differed from other axioms of set theory. For some time it was mainly used in the form of the Well-Ordering theorem (which is actually equivalent to the choice axiom). In this formulation, the axiom permits arguments by the so-called transfinite induction. For about the last 50 years it has usually been used in the form of the Maximal Principle of Zorn (published in the 1930s, though a version by Hausdorff was published earlier.)

If nontrivial variation with respect to some category S of abstract sets and arbitrary mappings is allowed in a category of variable sets, the axiom of choice tends not to hold as we have seen in Exercises 4.54, 6.12, and Section 10.3. In fact, the axiom is valid in certain very special toposes of variable sets determined relative to a category of abstract sets and arbitrary mappings, as was mentioned in Section 4.6.

Here we will demonstrate that Zorn's Maximal Principle is equivalent to the axiom of choice. We will show this for forms of the Maximal Principle that use both chains and directed (or filtered) posets. The latter form is more suitable for arguments made in mathematical practice, whereas it is the former that we will see as a direct consequence of the axiom of choice. In addition we will consider Hausdorff's Maximal Principle and some other consequences of the choice axiom. Our proofs of the maximal principles will follow directly from the famous Fixed- Point theorem of Bourbaki (B.15 below).

Axioms for Abstract Sets and Mappings

We have now seen most of the axioms we will require of the category S of abstract sets and mappings. As we progressed, some of the earlier axioms were included in later axioms. For example, the existence of the one-element set is part of the axiom that S has finite limits. Although we did not insist on it earlier, it is also the case that some of the axioms are more special than others. By this we mean that even though they hold in S they will not generally hold in categories of variable or cohesive sets. Thus, we are going to review the axioms here so that they can be considered all at once and grouped according to their generality.

The very first axiom, of course, is

AXIOM:S IS A CATEGORY

We have been emphasizing all along that the fundamental operation in a category, composition, is the basic tool for both describing and understanding all of the other properties of S.

The next group of three axioms is satisfied by any category of sets, variable or constant. In fact a category satisfying them is called a topos (in the elementary sense), and these categories have been studied intensively since 1969.