Question

Let Well $(A)$ be the set of all relations on $A$ which are well-orderings. If $r, r^{\prime} \in$ Well $(A)$, define $r \leqslant r^{\prime}$ iff $r \subseteq r^{\prime}$ and dom $[r]$ is an initial subset of the well-ordered set $\left(\operatorname{dom}\left[r^{\prime}\right], r^{\prime}\right)$. Show that $(\mathrm{Well}(A), \leqslant)$ is an inductively ordered set. Hence deduce the well-ordering property directly from Zorn's lemma.

    Let Well $(A)$ be the set of all relations on $A$ which are well-orderings. If $r, r^{\prime} \in$ Well $(A)$, define $r \leqslant r^{\prime}$ iff $r \subseteq r^{\prime}$ and dom $[r]$ is an initial subset of the well-ordered set $\left(\operatorname{dom}\left[r^{\prime}\right], r^{\prime}\right)$. Show that $(\mathrm{Well}(A), \leqslant)$ is an inductively ordered set. Hence deduce the well-ordering property directly from Zorn's lemma.

Set theory and its philosophy: a critical introduction

Michael Potter 1st Edition

Chapter 14, Problem 4

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Step 1

Let $C \subseteq \text{Well}(A)$ be a chain. We want to find an upper bound $u \in \text{Well}(A)$ such that $c \leqslant u$ for all $c \in C$. First, let's define the relation $u$ as the union of all relations in $C$. In other words, for any $(x, y) \in u$, Show more…

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Let Well $(A)$ be the set of all relations on $A$ which are well-orderings. If $r, r^{\prime} \in$ Well $(A)$, define $r \leqslant r^{\prime}$ iff $r \subseteq r^{\prime}$ and dom $[r]$ is an initial subset of the well-ordered set $\left(\operatorname{dom}\left[r^{\prime}\right], r^{\prime}\right)$. Show that $(\mathrm{Well}(A), \leqslant)$ is an inductively ordered set. Hence deduce the well-ordering property directly from Zorn's lemma.

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