# Category:Ordinals

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This category contains results about Ordinals.

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

$\alpha$ is an **ordinal** if and only if it fulfils the following conditions:

\((1)\) | $:$ | $\alpha$ is a transitive set | |||||||

\((2)\) | $:$ | $\Epsilon {\restriction_\alpha}$ strictly well-orders $\alpha$ |

where $\Epsilon {\restriction_\alpha}$ is the restriction of the epsilon relation to $\alpha$.

## Subcategories

This category has the following 26 subcategories, out of 26 total.

### C

- Counting Theorem (8 P)

### E

- Existence of Hartogs Number (3 P)

### F

- Finite Ordinals (7 P)

### H

- Hartogs' Lemma (Set Theory) (3 P)

### I

### L

### O

- Ordinal is Transitive (5 P)
- Ordinals are Well-Ordered (4 P)

### R

### S

### T

- Transfinite Induction (10 P)

### U

### Z

## 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
- Class such that Every Transitive Subset is Element of it Contains All Ordinals
- 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
- Counting Theorem

### E

- Element of Every Transitive-Closed Class is Ordinal
- Element of Ordinal is Ordinal
- Equality of Successors
- Equality of Successors implies Equality of Ordinals
- Equivalence of Definitions of Ordinal
- Existence of Disjoint Well-Ordered Sets Isomorphic to Ordinals
- Existence of Hartogs Number
- Exists Ordinal Greater than Set of Ordinals

### I

### M

### N

### O

- Order Isomorphism between Ordinals and Proper Class/Corollary
- Order Isomorphism between Ordinals and Proper Class/Lemma
- Ordering on Ordinal is Subset Relation
- Ordinal equals its Initial Segment
- Ordinal equals Successor of its Union
- Ordinal is Finite iff Natural Number
- Ordinal is Less than Successor
- Ordinal is not Element of Itself
- Ordinal is Proper Subset of Successor
- Ordinal is Subset of Successor
- Ordinal is Transitive
- Ordinal Membership is Asymmetric
- Ordinal Membership is Transitive
- 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

### S

- Set is Element of Successor
- Set is Ordinal iff Every Transitive Proper Subset is Element of it
- Set of Natural Numbers is Ordinal
- Set of Natural Numbers is Smallest Ordinal Greater than All Natural Numbers
- Strict Ordering of Ordinals is Equivalent to Membership Relation
- Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping
- Subset is Compatible with Ordinal Successor
- Subset of Ordinals has Minimal Element
- 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 Mapping on Ordinals is Strictly Progressing
- Successor of Element of Ordinal is Subset
- Successor of Ordinal Smaller than Limit Ordinal is also Smaller
- Successor Set of Ordinal is Ordinal
- Supremum Inequality for Ordinals