Definition:G-Ordered Class
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Definition
Let $A$ be a class.
Let $g$ be a progressing mapping.
Let $A$ be well-ordered by the subset relation such that:
\((1)\) | $:$ | the smallest element of $A$ is $\O$ | |||||||
\((2)\) | $:$ | every immediate successor $y$ is $\map g x$, where $x$ is the immediate predecessor of $y$ | |||||||
\((3)\) | $:$ | every limit element $x$ of $A$ is the union of the set of all predecessor elements of $x$ |
Then $A$ is said to be $g$-ordered.
Also see
- Results about $g$-ordered classes can be found here.
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 $\text {III}$ -- The existence of minimally superinductive classes: $\S 8$ Another characterization of $g$-sets