Definition:Backward Path (Class Theory)

Definition

Let $A$ be a class under a total ordering $\preccurlyeq$.

Let $S = \sequence {x_n}_{n \mathop \in \N}$ be a sequence in $A$ (either finite or infinite) such that:

$\forall i \in \N: x_i \in S \implies x_{i + 1} \prec x_i$

Then $S$ is referred to as a backward path of $\preccurlyeq$ (in $A$).

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering: Exercise $1.4$