# Category:Well-Orderings

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This category contains results about **Well-Orderings**.

Definitions specific to this category can be found in Definitions/Well-Orderings.

The ordering $\preceq$ is a **well-ordering** on $S$ if and only if *every* non-empty subset of $S$ has a smallest element under $\preceq$:

- $\forall T \subseteq S, T \ne \O: \exists a \in T: \forall x \in T: a \preceq x$

## Subcategories

This category has the following 20 subcategories, out of 20 total.

### C

- Counting Theorem (8 P)

### E

- Empty Set is Well-Ordered (3 P)

### F

### L

- Limit Elements (2 P)

### P

- Proper Well-Orderings (3 P)

### R

### S

### T

### U

### W

### Z

- Zermelo's Well-Ordering Theorem (12 P)

## Pages in category "Well-Orderings"

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

### C

- Canonical Order Well-Orders Ordered Pairs of Ordinals
- Categories of Elements under Well-Ordering
- Characterisation of Limit Element under Well-Ordering
- Characterization of Strictly Increasing Mapping on Woset
- Choice Function Exists for Set of Well-Ordered Sets
- Choice Function Exists for Well-Orderable Union of Sets
- Class which has Injection to Subclass of Well-Orderable Class is Well-Orderable
- Condition for Total Ordering to be Well-Ordering
- Conditions for Relation to be Well-Ordering
- Countable Set is Well-Orderable
- Counting Theorem

### D

### E

- Elements of Minimally Inductive Set are Well-Ordered
- Empty Class is Well-Ordered
- Empty Set is Well-Ordered
- Equality of Towers in Set
- Equality to Initial Segment Imposes Well-Ordering
- Equivalence of Definitions of Strict Well-Ordering
- Equivalence of Definitions of Strictly Well-Ordered Set
- Equivalence of Definitions of Well-Ordering
- Existence of Disjoint Well-Ordered Sets Isomorphic to Ordinals

### F

### I

### L

### M

### N

### O

- Order Automorphism on Well-Ordered Class is Forward Moving
- Order Automorphism on Well-Ordered Class is Identity Mapping
- Order Isomorphism between Wosets is Unique
- Order Isomorphism from Woset onto Subset
- Order Isomorphism on Well-Ordered Set preserves Well-Ordering
- Order-Preserving Bijection on Wosets is Order Isomorphism
- Ordering where every Subclass has Smallest Element is Well-Ordering

### P

### R

### S

- Sandwich Principle
- Second Principle of Transfinite Induction
- Set of Integers Bounded Above has Greatest Element
- Set of Integers Bounded Below has Smallest Element
- Set of Integers can be Well-Ordered
- Set of Integers is not Well-Ordered by Usual Ordering
- Set of Non-Negative Real Numbers is not Well-Ordered by Usual Ordering
- Set with Choice Function is Well-Orderable
- Set with Slowly Progressing Mapping on Power Set with Self as Fixed Point is Well-Orderable
- Slow g-Tower is Slowly Well-Ordered under Subset Relation
- Strict Lower Closure of Limit Element is Infinite
- Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping
- Strictly Increasing Mapping Between Wosets Implies Order Isomorphism
- Subclass of Well-Ordered Class is Well-Ordered
- Subset of Well-Ordered Set is Well-Ordered

### T

### U

### W

- Well-Orderable Set has Choice Function
- Well-Ordered Class is not Isomorphic to Initial Segment
- Well-Ordered Classes are Isomorphic at Most Uniquely
- Well-Ordered Induction
- Well-Ordered Transitive Subset is Equal or Equal to Initial Segment
- Well-Ordering is not necessarily Usual Ordering
- Well-Ordering Minimal Elements are Unique
- Well-Ordering of Class of All Ordinals under Subset Relation
- Well-Ordering on Class is not necessarily Proper
- Well-Ordering Principle
- Woset is Isomorphic to Set of its Initial Segments
- Wosets are Isomorphic to Each Other or Initial Segments