172 V I I I : Set theory as a foundation for mathematics This material is basically supplementary, and it was not covered in the course. In the first section we discuss the basic axioms of set theory and the desirability of making the axiom system as simple and irredundant as possible. The main objective of the second section is to describe exactly how one can simplify our assumptions for set theory, with particular attention to our fairly lengthy set of axioms for number systems; it turns out that one can replace these by a single assumption that is far more concise and is also central to the basic logical consistency issues raised in the previous unit. In the third section we prove results stated in Unit V about the essential uniqueness of number systems satisfying our axioms for the integers and the real number system. The fourth and final section covers a topic that fits in with both the naïve and formal approaches. In Unit I of these notes we mentioned that the axioms for Euclidean geometry were viewed as a major portion of the logical foundations for mathematics up to the early 19th century, and that by the end of that century set theory was quickly evolving into a new logical basis for the subject. One natural question is whether the axioms for classical Euclidean geometry can be integrated into the new framework for mathematics, and if so the next question is how this can be done. In the final section we explain how one can view the classical axiomatic approach to geometry within the environment of set theory. V I I I. 1 : Formal development of set theory (Halmos, §§ 1 – 10, 14; Lipschutz, § 1.12) In Section I I.1 we began by describing set theory from a naïve viewpoint and then indicated how one could set things up more formally. In most of the notes, our approach has been very much on the naïve side; usually we have introduced assumptions about set theory as they were needed to continue or expedite the discussion without worrying too much about how one should express everything in a completely rigorous manner. This allowed us to develop the subject fairly rapidly. At some points we mentioned the need to be more specific about some issues (e.g., describing the “admissible” logical statements that can be used to describe sets) or the possibility of deriving some of our assumptions as logical consequences of the others. For example, in Section I I I.2 of the notes we mentioned that the existence of objects with the properties of ordered pairs can be proved from the other assumptions; details appear on pages 23 – 25 of Halmos. Frequently the proofs of such implications are somewhat complicated and unmotivated and the approach may seem artificial, and therefore we have simply added assumptions in Section I I I.2 and elsewhere to save time and to focus attention on points that are directly related to the uses of set theory in the mathematical sciences.173 However, once the basics of set theory have been covered and assimilated, there are some extremely compelling reasons to look back and examine the assumptions in order to see if they can be simplified and redundant assumptions can be eliminated. One major reason to look for simpler and more concise assumptions is a basic principle in the philosophy of science called Ockham’s razor, which was originally stated by William of Ockham (1285 – 1349). In modern language, this principle states that complications should not be introduced unless they are necessary or in more imperative terms do not invent unnecessary entities to explain something. Since we shall appeal to Ockham’s razor at other points in this unit, we include an online reference to a biography for William of Ockham: http://plato.stanford.edu/entries/ockham/ In the mathematical sciences there are important practical justifications for using Ockham’s razor that go well beyond simplicity of exposition. Since the mathematical sciences are so heavily dependent upon deductive logic, it is absolutely essential to have some assurance that the basic assumptions are logically sound. If the assumptions for some theory lead to logical contradictions, serious questions arise about the validity and reliability of the theory’s conclusions and value. Simplified lists of basic assumptions turn out to be extremely useful for testing the logical soundness of a mathematical system. The reason is obvious; there are fewer things to verify, for much of the work is redirected into verifying the original assumptions are equivalent to the simplified ones. The advantages of simplified lists of assumptions are also illustrated very clearly by examples within mathematics itself. In mathematical proofs by contradiction, the underlying idea for proving P implies Q is to assume that P is true, to add an assumption that Q is false, and to use the new, longer set of hypotheses to obtain a contradiction. This method has a fundamental implication: As lists of assumptions become longer and more complicated, one must be increasingly careful in checking whether the entire list of assumptions is logically consistent. It is generally much easier to check shorter systems of axioms for consistency than it is to check longer ones, so if we want to understand the consistency properties of our axioms it is highly desirable to have an equivalent version which is as simple as possible. Summary of the basic axioms As noted in Unit VI I, one standard axiomatic approach to set theory in present day mathematics is based upon axioms introduced by E. Zermelo during the first decade of the 20th century, with a few subsequent modifications due to other mathematicians, most notably A. Fraenkel. Versions of most Zermelo – Fraenkel (ZF) axioms have been introduced in previous units, and all the other assumptions we have introduced turn out to be consequences of these axioms, all of which are listed below: • The Axiom of Extensionality (see Section I I.1) • The Axiom of Pairs (see Section I I.2 and also below)174• The Axiom of Specification (see Section I I.2) • The Axiom of the Power Set (see Section I I I.3) • The Axiom of Unions (see Section I I I.3) • The Axiom of Replacement (see Section I V.4) • The Axiom of Foundation (see Section I I I.5) • The Axiom of Number Systems (see Sections V.1 and V.4 as well as the next paragraph) Note that the Axiom of Choice is missing from this list; if this is added, one