Problem 2 - A Course in Galois Theory

Question

A total order on a set \(S\) is said to be a 'well-ordering' if every nonempty subset of \(S\) has a least element. Use Zorn's lemma to show that every non- empty set can be given a well-ordering. (Hint: Define a partial order \(\prec\) on all pairs \((T, \leqslant)\), where \(T\) is a subset of \(S\), and \(\leqslant\) is a well-ordering on \(T\), by saying that $$ \left(T_{1}, \leqslant_{1}\right) \prec\left(T_{2}, \leqslant_{2}\right) $$ if first \(T_{1} \subseteq T_{2}\), secondly the orderings \(\leqslant_{1}\) and \(\leqslant_{2}\) coincide on \(T_{1}\) and thirdly $$ P_{t}=\left\\{x \in T_{2}: x \leqslant_{2} t\right\\} $$ is contained in \(T_{1}\) whenever \(t\) is in \(T_{1}\). Apply Zorn's lemma to this.)

Step-by-Step Solution

Answer

Every non-empty set can be well-ordered by using Zorn's Lemma to find a maximal well-ordering relation on subsets of the set.

Step 1: Understand the Goal

Our goal is to prove that every non-empty set \(S\) can be well-ordered, meaning every subset of \(S\) has a least element.

Step 2: Define Partial Order

Define a partially ordered set consisting of pairs \((T, \leqslant)\), where \(T\) is a subset of \(S\), and \(\leqslant\) is a well-ordering of \(T\). These pairs are ordered by \( \prec \) such that \((T_1, \leqslant_1 ) \prec (T_2, \leqslant_2 )\) means: (1) \(T_1 \subseteq T_2\), (2) \(\leqslant_1\) and \(\leqslant_2\) coincide on \(T_1\), and (3) for every \(t \in T_1\), the initial segment \( P_t = \{x \in T_2 : x \leqslant_2 t \}\) is contained in \(T_1\).

Step 3: Check Chain Conditions

Consider a chain (i.e., a totally ordered subset) \( \{(T_i, \leqslant_i)\}_i \) under the relation \( \prec \). Take the union of all \(T_i\) in this chain, denoted \(T^* = \cup T_i\). Define \(\leq^*\) by agreeing with \(\leqslant_i\) on each \(T_i\). This is a well-ordering on \(T^*\) because it maintains the well-ordering properties on every subset \(T_i\) in the chain.

Step 4: Apply Zorn's Lemma

Zorn's Lemma states that if every chain in a partially ordered set has an upper bound in the set, then there is a maximal element. In this context, given our partial order \(\prec\), the pair \((T^*, \leq^*)\) acts as an upper bound for any chain in our partially ordered set. So, by Zorn's Lemma, there exists a maximal pair \((T_{max}, \leqslant_{max})\).

Step 5: Conclude Well-Ordering

By construction, if \(T_{max}\) is a proper subset of \(S\), one could extend \((T_{max}, \leqslant_{max})\) by adding an element from \(S \setminus T_{max}\), contradicting maximality. Hence, \(T_{max} = S\), and \(\leqslant_{max}\) is a well-ordering on \(S\), as desired.

Key Concepts

Well-Ordering

A well-ordering is a special type of ordering on a set. It means that every nonempty subset has a least (or smallest) element.
In more concrete terms, imagine you have a collection of numbers, like positive integers. In a well-ordered set, no matter which group of numbers you pick, you can always point to the smallest one.
Well-ordering is crucial because it helps in organizing elements in a predictable pattern. This concept becomes handy in many areas of mathematics since it makes sure that step-by-step processes can be carried out in an orderly fashion without infinite loops.

Partial Order

A partial order is a way of arranging some, but not necessarily all, elements of a set into a sequence. It's not as strict as a total order because it doesn't require every pair of elements to be comparable.
To formally define it, a relation \( \prec \) on a set is a partial order if it satisfies three conditions:

• Reflexivity: every element is related to itself.
• Antisymmetry: if one element is related to another and vice versa, they are equal.
• Transitivity: if one element is related to a second, and the second is related to a third, the first is related to the third.

Partial orders appear in everyday situations, like when organizing tasks that depend on each other, where some order of tasks is possible, but not every task relates to every other one.

Maximal Element

A maximal element in a set is one that cannot be extended or enlarged while retaining the properties of the set.
In a partially ordered set, a maximal element is not necessarily the greatest of all; it's simply one that you can't "add" anything to without breaking the rules of the order.
Think of a maximal element like a frontier beyond which one can't extend. In the context of using Zorn's Lemma, finding a maximal element ensures we are reaching a stage where no further additions or extensions are possible, hence arriving at a complete set or solution.

Set Theory

Set theory is the branch of mathematical logic that studies collections of objects, known as sets. It forms the groundwork for a large part of modern mathematics.
In set theory, sets can be numbers, symbols, or even other sets, and they are fundamental to understanding mathematical structures.
Key concepts of set theory include:

• Membership: how elements belong to a set.
• Subset: when all elements of one set are contained in another.
• Union and Intersection: operations that combine sets.
• Power Set: the set of all subsets of a set.

In advanced mathematics, set theory helps define structures and properties systematically, supporting much of algebra, geometry, and mathematical analysis.