# 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 7 subcategories, out of 7 total.

### S

### W

## Pages in category "Well-Orderings"

The following 55 pages are in this category, out of 55 total.

### A

### E

- Elements of Minimal Infinite Successor Set are Well-Ordered
- Equality of Towers in Set
- Equality to Initial Segment Imposes Well-Ordering
- Equivalence of Definitions of Strictly Well-Ordered Set
- Equivalence of Definitions of Well-Ordering
- Equivalence of Definitions of Well-Ordering/Definition 1 implies Definition 2

### F

### I

### L

### M

### N

### O

### P

### R

### S

- Sandwich Principle
- 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
- Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping
- Strictly Increasing Mapping Between Wosets Implies Order Isomorphism
- Subset of Well-Ordered Set is Well-Ordered

### T

### W

- Well-Ordered Induction
- Well-Ordered Transitive Subset is Equal or Equal to Initial Segment
- Well-Ordering Minimal Elements are Unique
- Well-Ordering Principle
- Well-Ordering Theorem implies Hausdorff Maximal Principle
- Woset is Isomorphic to Set of its Initial Segments
- Wosets are Isomorphic to Each Other or Initial Segments