Category:Properties of Class of All Ordinals
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This category contains pages concerning Properties of Class of All Ordinals:
Let $\On$ denote the class of all ordinals.
$\On$ has the following properties:
Zero is an Ordinal
The natural number $0$ is an element of $\On$.
Successor of Ordinal is an Ordinal
Let $\On$ denote the class of all ordinals.
Let $\alpha \in \On$ be an ordinal.
Then its successor set $\alpha^+ = \alpha \cup \set \alpha$ is also an ordinal.
Union of a Chain of Ordinals is an Ordinal
Let $C$ be a chain of elements of $\On$.
Then its union $\bigcup C$ is also an element of $\On$.
Superinduction Principle
Let $A$ be a class which satisfies the following $3$ conditions:
\((1)\) | $:$ | $A$ contains the zero ordinal $0$: | \(\ds 0 \in A \) | ||||||
\((2)\) | $:$ | $A$ is closed under successor mapping: | \(\ds \forall \alpha:\) | \(\ds \paren {\alpha \in A \implies \alpha^+ \in A} \) | |||||
\((3)\) | $:$ | $A$ is closed under chain unions: | \(\ds \forall C:\) | \(\ds \bigcup C \in A \) | where $C$ is a chain of elements of $A$ |
That is, let $A$ be a superinductive class under the successor mapping.
Then $A$ contains all ordinals:
- $\On \subseteq A$
Subcategories
This category has only the following subcategory.
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Pages in category "Properties of Class of All Ordinals"
The following 8 pages are in this category, out of 8 total.