A Century Ago

Friday of last week was Mr. and Mrs. John Feeney’s silver wedding anniversary which they quietly observed until evening, when a group of friends gave them a very pleasant surprise party. Refreshments and a merry social time were enjoyed and some choice gifts of silver, etc., were presented. The party dispersed at a seasonable hour, wishing their host and hostess many happy returns of the day. Mr. and Mrs. D. L. Greig and Mr. and Mrs. J. W. Fletcher were instrumental in arranging this pleasant affair.

paper, Zermelo uses the "Axiom of Choice" to prove that every set can be wellordered (to be explained shortly). However, the mathematical community reacted negatively to Zermelo's paper and he was motivated to respond in his first 1908 paper.

Zermelo's Axioms of Set Theory
Note. In the second of these two papers, Zermelo presents the first axiomatic set theory. In that era, set theory was undergoing a bit of turmoil. Russell's Paradox had been revealed and the foundations of set theory were coming into question. Part of Zermelo's paper involves constructions of sets and his approach does not allow the constructions of sets which are "too big" (Russell, at roughly the same time, introduces his "theory of types" to address this problem). In addition, Zermelo seems to be the first to see that the existence of infinite sets must be taken as an axiom (page 199 of van Heijenoort).
Note. Zermelo states 7 axioms (to paraphrase):   "If T is a set whose elements all are sets that are different from ∅ and mutually disjoint, its union ∪ T includes at least one subset S 1 having one and only one element in common with each element of T ." Note. The Axiom of Choice can be used to construct a (in the sense of Lebesgue) nonmeasurable set. A related result is the Banach-Tarski paradox which uses the Axiom of Choice to decompose a sphere into five pieces, two of which can be combined (through rigid rotations) to give a copy of the sphere and the other three which can be (rigidly) combined to give a second copy of the sphere. That is, the sphere can be cut into pieces which can then be combined to make two copies of the sphere, thus doubling the volume of the sphere out of nothing! Note. We now consider 3 "principles" equivalent to the Axiom of Choice:

Well-Ordering Principle (Zermelo's Theorem).
Every set can be well-ordered.

Maximal Principle I (Zorn's Lemma).
Let (P, <) be a nonempty partially ordered set and let every chain in P have an upper bound. Then P has a maximal element.

Maximal Principle II (Tukey's Lemma).
Let F be a nonempty family of sets. If F has finite character, then F has a maximal element (maximal with respect to set inclusion).
We now elaborate on these ideas.

Well-Orderings
Definition. [Hrbacek and Jech, 1984] A set R is a binary relation on set A if all elements of R are ordered pairs of elements of A. We denote (a, b) ∈ R as aRb. A binary relation R in A which is reflexive (for all a ∈ A, aRa), antisymmetric (for all a, b ∈ A with aRb and bRa, we have a = b), and transitive (for all a, b, c ∈ A with aRb and bRc, we have aRc) is said to be a partial ordering of A. We denote such an ordering as ≤ . If for a, b ∈ A, either a ≤ b or b ≤ a then a and b are comparable. An ordering ≤ of A is a total ordering (or linear ordering) if any two elements of A are comparable.

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Example. Subset inclusion ⊂ is a partial ordering on, say, P(R). Certain sets are not comparable under this partial ordering (for example, (1,3) and (0, 2)), so this is not a total ordering. Example. Less than or equal to ≤ is a total ordering on R.
Definition. A total ordering ≺ of a set A is a well-ordering if every nonempty subset of A has a ≺-least element.
Example. Less than or equal to ≤ is a well-ordering on N. However, ≤ is not a well-ordering of R (consider the nonempty subset A = (0, 1) of R).
Note. The Well-Ordering Principle says that there is a well-ordering of every set.
Note. You may have heard that there is no ordering of the complex numbers.
What this means is that there is no way to define "≤" on C in such a way that the usual ordering on R comes as a "subordering." This idea of well-ordering a set should not be confused with ordering of a field (in, for example, the definition of R as a "complete ordered field").

Chains
Definition. A subset C of a partially ordered set (P, ≺) is a chain in P if C is totally ordered by ≺. u ∈ P is an upper bound of chain C if c ≺ u for all c ∈ C. a ∈ P is a maximal element if a ≺ x for no x ∈ P .
Axiom of Choice 7 Note. The Maximal Principle I (Zorn's Lemma) states that a nonempty partially ordered set P for which every chain in P has an upper bound, has a maximal element.

Finite Character
Definition. Let F be a family of sets. We say that F has finite character if for each set X, X ∈ F if and only if every finite subset of X belongs to F.
Note. The Maximal Principle II (Tukey's Lemma) states that every nonempty family of sets of finite character has a maximal element (maximal with respect to subset inclusion).