# Category:Ordinals

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This category contains results about Ordinals.

Definitions specific to this category can be found in Definitions/Ordinals.

Let $S$ be a set.

Let $\Epsilon \! \restriction_S$ be the restriction of the epsilon relation on $S$.

Then $S$ is an **ordinal** if and only if:

- $S$ is a transitive set
- $\Epsilon \! \restriction_S$ strictly well-orders $S$.

## Subcategories

This category has the following 9 subcategories, out of 9 total.

### E

### F

### H

### I

### O

### S

## Pages in category "Ordinals"

The following 97 pages are in this category, out of 97 total.

### C

- Canonical Order Well-Orders Ordered Pairs of Ordinals
- Cardinal Inequality implies Ordinal Inequality
- Cardinal Number Less than Ordinal
- Cardinal Number Less than Ordinal/Corollary
- Cardinal Number Plus One Less than Cardinal Product
- Cofinal Limit Ordinals
- Cofinal Ordinal Relation is Reflexive
- Cofinal Ordinal Relation is Transitive
- Cofinal to Zero iff Ordinal is Zero
- Condition for Cofinal Nonlimit Ordinals
- Condition for Woset to be Isomorphic to Ordinal
- Copi's Identity

### E

### I

### L

### M

### N

### O

- Order Isomorphism between Ordinals and Proper Class/Lemma
- Ordering on Ordinal is Subset Relation
- Ordinal is Finite iff Natural Number
- Ordinal is Less than Successor
- Ordinal is not Element of Itself
- Ordinal is Subset of Successor
- Ordinal is Transitive
- Ordinal Membership is Asymmetric
- Ordinal Membership is Trichotomy
- Ordinal Subset is Well-Ordered
- Ordinal Subset of Ordinal is Initial Segment
- Ordinals are Totally Ordered
- Ordinals are Well-Ordered
- Ordinals are Well-Ordered/Corollary
- Ordinals Isomorphic to the Same Well-Ordered Set

### R

### S

- Set is Element of Successor
- Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping
- Subset is Compatible with Ordinal Successor
- Subset of Ordinals has Minimal Element
- Successor in Limit Ordinal
- Successor is Less than Successor
- Successor is Less than Successor/Sufficient Condition
- Successor is Less than Successor/Sufficient Condition/Proof 1
- Successor is Less than Successor/Sufficient Condition/Proof 2
- Successor of Element of Ordinal is Subset
- Successor Set of Ordinal is Ordinal
- Supremum Inequality for Ordinals

### T

- Transfinite Induction
- Transfinite Induction/Principle 1
- Transfinite Induction/Principle 1/Proof 2
- Transfinite Induction/Principle 2
- Transfinite Induction/Schema 1
- Transfinite Induction/Schema 1/Proof 1
- Transfinite Induction/Schema 1/Proof 2
- Transfinite Induction/Schema 2
- Transfinite Induction/Schema 2/Proof 1
- Transfinite Induction/Schema 2/Proof 2
- Transfinite Recursion
- Transfinite Recursion/Corollary
- Transfinite Recursion/Theorem 1
- Transfinite Recursion/Theorem 2
- Transfinite Recursion/Uniqueness of Transfinite Recursion
- Transitive Set is Proper Subset of Ordinal iff Element of Ordinal
- Transitive Set is Proper Subset of Ordinal iff Element of Ordinal/Corollary