Exercise 20, § 2

A set X is said to be transitive if the relation $x\in X$ implies $x\subset X$.

(a) If Y is a transitive set, then so is $Y\cup\{Y\}$. If $(Y_\iota)_{\iota\in I}$ is a family of transitive sets, then $\bigcup_{\iota\in I}Y_\iota$ and $\bigcap_{\iota\in I}Y_\iota$ are transitive.

(b) A set X is a pseudo-ordinal if every transitive set Y such that $Y\subset X$ and $Y\neq X$ is an element of X. A set S is said to be decent if the relation $x\in S$ implies $x\notin x$. Show that every pseudo-ordinal is transitive and decent (consider the union of the decent transitive subsets of X, and use (a)). If X is a pseudo-ordinal, then so is $X\cup\{X\}$.

(c) Let X be a transitive set, and suppose that each $x\in X$ is a pseudo-ordinal. Then X is a pseudo-ordinal (note that, for each $x\in X$, $x\cup\{x\}$ is a pseudo-ordinal contained in X).

(d) Show that $\varnothing$ is a pseudo-ordinal and that every element of a pseudo-ordinal X is a pseudo-ordinal. (Consider the union of the transitive subsets of X whose elements are pseudo-ordinals.)

(e) If $(X_\iota)_{\iota\in I}$ is a family of pseudo-ordinals, then $\bigcap_{\iota\in I}X_\iota$ is the least element of this family (with respect to the relation of inclusion). (Use (b).) Deduce that, if E is a pseudo-ordinal, the relation $x\subset y$ between elements $x,y$ of E is a well-ordering relation.

(f) Show that for each ordinal $\alpha$ there exists a unique pseudo-ordinal $\mathrm{E}_\alpha$ such that $\mathrm{Ord}(\mathrm{E}_\alpha) = \alpha$ (use (e) and criterion C60). In particular, the pseudo-ordinals whose order-types are $0, 1, 2 = 1 + 1$, and $3 = 2 + 1$ are respectively