Set theory (Math 320)

Spring semester, 2008/9

Text

The main text for this course is the lectures. There is also a supplementary text, called Sets, Classes, and Families: Notes on set theory, obtainable in two ways:

from the METU Library copy center, or
as a download from here (it's the file <set-theory.*>).

This text is a thorough revision of the text used last time I taught this course. The preface mentions some useful books; and there are many others.

Again though, the main text is the lectures. In the lectures, I shall not cover everything in the printed notes; this depends in part on the interest of the class. Moreover, the lectures need not have the same order as the printed notes. Finally, I cannot guarantee that everything that will be in the lectures is covered in the notes (although this was my intention in writing the notes).

Lectures

Tuesdays, 12.40–13.30, in M-103
Thursdays, 10.40–12.30, in M-104

We start with §§ 6.3–6 of the typeset notes; but we use Theorems 6.3.4 and 6.4.3 as definitions. [More precisely, modified forms of 6.3.4 and 6.4.3 are definitions, so that Lemma 6.5.4 will be true.] With this understanding (and using α+1 as α′) we can do Exercises 6.9–20.

Then we go back to Chapter 2, although not in every detail. (Chapter 1 can be read on one's own.) Exercises 2.1–8 can be done. [However, Exercise 2.5(ii) should really be in Chapter 3.] The first exam comes here.

Skipping §§ 2.5–6 for now, we continue to Chapter 3 and cover things roughly in the order of the notes. We supplement §4.5 by constructing a countably infinite class using the idea behind the Foundation Axiom mentioned in ¶4.5.4. The second exam comes here.

Because of the possibility just mentioned of constructing a countably infinite class, we do not need §4.7 (so then Exercise 4.21 can be ignored).

We skip ¶¶5.2.3–4, though not the argument in ¶5.2.7. This argument then justifies ¶5.2.5 in case 𝑨 is a set. Similar considerations hold in §5.3. But all of the exercises in Ch. 5 are still doable.

In ¶6.1.3, an alternative definition of ordinal is possible: an ordinal is a transitive set on which membership is a transitive relation. However, for this definition to yield the desired properties, we must introduce the Foundation Axiom (¶8.2.1), which we state in the form

∀𝑥 ∃𝑦 (𝑥 ≠ ∅ ⇒ (𝑦 ∈ 𝑥 & 𝑥 ∩ 𝑦 = ∅)).

From this it can be shown that ordinals and the class of ordinals are indeed well-ordered by membership.

We skip § 7.1. In ¶7.2.1, the definition of cardinality for sets that cannot be well-ordered is of little importance, since we shall assume the Axiom of Choice, from which we obtain that every set can be well-ordered (§ 7.5). In ¶ 7.7.2, the definition of multiplication in ℝ needs to be corrected to take into account the negative numbers in ℚ.

We do not cover Chapter 8.

There will be no class on May 21 (or May 19 of course) because of Antalya Algebra Days.

Examinations

There will be three exams in term (90 minutes each), along with the final exam (2 hours). The exams in term will be in the evening, at 5:40 p.m. (that is, 17.40). I have scheduled them on Mondays because I do not want to give exams on days when we have class. For the course, the best two in-term exam scores will count 30% each; the final exam will count 40%. There will be no make-up exams; instead, everybody can miss one in-term exam, for whatever reason.

March 16, in M-05. Solutions. Scores.
April 6, in M-102. Solutions. Scores.
May 4, in M-103. Solutions. Scores. (Note added May 14, 2009: there were some mistakes in an earlier report of these scores.)
Final exam, Friday, June 5, at 16.30, in P5. Solutions. Scores.

General remarks about set theory

I recall here the catalogue-description of the course:

Language and axioms of set theory. Ordered pairs, relations and functions. Order relation and well ordered sets. Ordinal numbers, transfinite induction, arithmetic of ordinal numbers. Cardinality and arithmetic of cardinal numbers. Axiom of choice, generalized continuum hypothesis.

Set-theory is useful to mathematics in the following way. We establish certain properties of sets, usually by means of axioms, such as the so-called Zermelo–Fraenkel system of axioms. Then we can define various standard mathematical structures (such as the ordered field ℝ of real numbers), and we can prove the various properties of these structures, without having to introduce new assumptions. In this way, set-theory provides a uniform foundation for mathematics.

Set-theory is also worth pursuing for its own sake. The development of set-theory involves the creation of certain new structures (such as the class ON of ordinal numbers, or the class L of constructible sets) that are interesting in themselves. Set-theory gives us certain useful techniques (such as various forms of proof by induction and definition by recursion).

Number-theory contains theorems (such as the so-called Fermat's Last Theorem) that are easy to understand, but almost impossibly difficult to prove. Some theorems of set-theory are harder to understand (and also difficult to prove), but are mind-blowing in their connotations. For example, ℝ is strictly greater in cardinality than the set ω of natural numbers; in a word, ℝ is uncountable; however, the Zermelo–Fraenkel axioms do not determine whether ℝ has the least uncountable cardinality.