# Prove Hausdorff's Maximality Principle by the W.O.P.

• Terrell
In summary, the conversation discusses the proof of Hausdorff's Maximality Principle using the Well-Ordering Principle. Three cases are considered and it is shown that the Well-Ordering Principle implies the Axiom of Choice, which in turn implies the Hausdorff's Maximality Principle. Alternative case 3 is also proposed, but it is unclear if it uses the Well-Ordering Principle and Axiom of Choice correctly. The use of the principle of transfinite recursive definition is suggested for future consideration in proving this statement.
Terrell

## Homework Statement

Show Hausdorff's Maximality Principle is true by the Well-Ordering Principle.

2. Relevant propositions/axioms

## The Attempt at a Solution

Case 1: ##\forall x,y\in X## neither ##x\prec y## or ##y\prec x## is true. Hence any singleton subset of ##X## is a maximal linear order.

Case 2: ##\prec## is a linear order on ##X##, then ##X## is itself the maximal linearly ordered subset of ##X##.

Case 3: ##\prec## is undefined on some pairs ##a,b\in X##. Then ##\prec## is not the relation that well-orders ##X## since there exist subsets of ##X## that does not have a least element.
Question: How can I use the Well-Ordering Principle here? I was thinking of defining a new relation ##\prec_w## that well-orders a finite ##W\subseteq X##, but I'm thinking that that cannot be right since ##\prec_w## is not an arbitrary partial-order relation in ##X##. Which part of this problem should I think more about in order to help me proceed? Or May I simply say that we remove all elements in ##X## that has an undefined relation with all other elements in ##X##?

EDIT: (Alternative case 3: W.O.P.##\rightarrow## A.C. ##\rightarrow## H.M.P.)
Suppose ##\{X_{\lambda}\}_{\lambda\in\Lambda}\neq\emptyset## and ##\forall\lambda\in\Lambda, X_{\lambda}\neq\emptyset##. We want to show ##\prod_{\lambda\in\Lambda}X_{\lambda}\neq\emptyset##. By the well-ordering principle, ##\exists\prec_{x}\forall\lambda\in\Lambda##, ##X_\lambda## is well-ordered; that is with respect to ##\prec_x##, ##\forall\lambda\in\Lambda\forall W\subseteq X_{\lambda}##, ##W## has a least element. Hence, we can simply define a choice-function ##f: \{X_\lambda\}_{\lambda\in\Lambda}\rightarrow\bigcup_{\lambda\in\Lambda}X_\lambda## such that ##f(X_{\lambda})=x_{0}^{\lambda}## denote the least element in ##X_{\lambda}##. Therefore, ##\prod_{\lambda\in\Lambda}X_{\lambda}\neq\emptyset##.

Finally, we want to show A.C. ##\rightarrow## H.M.P. Suppose ##\prec## does not well-order ##X##, then ##\exists a,b\in X## where neither ##a\prec b## or ##b\prec a## are true. Consider some partition of ##X## denoted ##P[X]##. Define ##\hat{X}:=\{\hat{x}\in X:\forall\overline{X}\in P[X]\forall x\in\overline{X},\neg (\hat{x}\prec x\quad\lor\quad x\prec\hat{x})\}##. By the axiom of choice, ##\hat{X}\neq\emptyset##. Thus, ##X^*=X\setminus\hat{X}## is linearly ordered such that ##\forall x\in\hat{X}, X^*\cup\{x\}## is no longer linearly ordered. Therefore, ##X^*\subseteq X## is a maximal linearly ordered subset of ##X##.

So I think I have two questions. First, is how to proceed directly from W.O.P. to H.M.P. and second, is I would like a verification on my alternative case 3 (i.e. did I use W.O.P. and A.C. correctly?)

#### Attachments

• WOP HMP AOC.png
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I can't follow your reasoning in the last paragraph. It does not seem to me that AC implies that ##\hat X\neq \emptyset##. Consider the case X={a,b,c,d} with the only order relations being a>b and c>d. Then X is not well-ordered because the subset {b,c} has no first element. Also, neither b<c nor c<b is true. But every element of X is in one order relation, so no element can be in ##\hat X##, hence ##\hat X=\emptyset##.

Also, is the principle of transfinite induction or the principle of transfinite recursive definition in your toolkit? They can sometimes be useful in making proofs of things like this.

Terrell
andrewkirk said:
the principle of transfinite recursive definition in your toolkit?
I'll consider using this when I get my mind back to this problem. Thanks for the hint! :)

## 1. What is Hausdorff's Maximality Principle?

Hausdorff's Maximality Principle is a mathematical concept that states that for any set of objects, there exists a maximal element that is not contained within any other element in the set. This principle is often used in set theory and has applications in other areas of mathematics, such as topology and functional analysis.

## 2. What is the W.O.P. and how does it relate to Hausdorff's Maximality Principle?

The W.O.P., or Well-Ordering Principle, is a fundamental axiom of set theory that states that every non-empty set can be well-ordered. This means that for any set, there exists a way to arrange its elements in a particular order. Hausdorff's Maximality Principle can be proven using the W.O.P. by showing that every set of objects has a maximal element, which can be obtained by well-ordering the set.

## 3. How is Hausdorff's Maximality Principle used in mathematics?

Hausdorff's Maximality Principle is used in various areas of mathematics, such as set theory, topology, and functional analysis. In set theory, it is used to prove the existence of certain types of sets, such as well-ordered sets and maximal elements. In topology, it is used to define compact spaces, and in functional analysis, it is used to prove the existence of certain types of functions, such as continuous functions.

## 4. Can Hausdorff's Maximality Principle be proven using other methods besides the W.O.P.?

Yes, Hausdorff's Maximality Principle can be proven using other methods, such as Zorn's Lemma or the Axiom of Choice. However, the W.O.P. is often the most straightforward and intuitive method for proving this principle.

## 5. What are some real-world applications of Hausdorff's Maximality Principle?

Hausdorff's Maximality Principle has applications in various fields, such as economics, computer science, and game theory. In economics, it is used to model decision-making processes, and in computer science, it is used to optimize algorithms and data structures. In game theory, it is used to analyze strategic interactions and find optimal solutions.

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