# Category:Cardinals

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This category contains results about **Cardinals**.

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

Let $S$ be a set.

Associated with $S$ there exists a set $\map \Card S$ called the **cardinal of $S$**.

It has the properties:

- $(1): \quad \map \Card S \sim S$

that is, $\map \Card S$ is (set) equivalent to $S$

- $(2): \quad S \sim T \iff \map \Card S = \map \Card T$

## Subcategories

This category has the following 4 subcategories, out of 4 total.

## Pages in category "Cardinals"

The following 63 pages are in this category, out of 63 total.

### C

- Cantor-Bernstein-Schröder Theorem
- Cardinal Equal to Collection of All Dominated Ordinals
- Cardinal Inequality implies Ordinal Inequality
- Cardinal Number Equivalence or Equal to Universe
- Cardinal Number is Ordinal
- Cardinal Number Less than Ordinal
- Cardinal Number Less than Ordinal/Corollary
- Cardinal Number Plus One Less than Cardinal Product
- Cardinal of Cardinal Equal to Cardinal
- Cardinal of Cardinal Equal to Cardinal/Corollary
- Cardinal of Finite Ordinal
- Cardinal of Union Equal to Maximum
- Cardinal of Union Less than Cardinal of Cartesian Product
- Cardinal One is Cancellable for Cardinal Sum
- Cardinal Product Distributes over Cardinal Sum
- Cardinal Product Equal to Maximum
- Cardinal Product Equinumerous to Ordinal Product
- Cardinal Zero is Less than Cardinal One
- Cardinalities form Inequality implies Difference is Nonempty
- Cardinality of Image of Mapping of Intersections is not greater than Weight of Space
- Cardinality of Image of Set not greater than Cardinality of Set
- Cardinality of Power Set is Invariant
- Cardinality of Power Set of Natural Numbers Equals Cardinality of Real Numbers
- Cardinality of Set is Finite iff Set is Finite
- Cardinality of Set less than Cardinality of Power Set
- Cardinality of Set of Singletons
- Cardinality of Singleton
- Cardinality of Union not greater than Product
- Cardinals are Totally Ordered
- Cardinals form Equivalence Classes
- Condition for Cartesian Product Equivalent to Associated Cardinal Number
- Condition for Existence of Cardinal Sum
- Condition for Set Equivalent to Cardinal Number
- Condition for Set Union Equivalent to Associated Cardinal Number

### F

### I

### N

### O

### P

### S

- Set Less than Cardinal Product
- Set of All Mappings is Small Class
- Set of All Mappings of Cartesian Product
- Set of Cardinality not Greater than Cardinality of Finite Set is Finite
- Subset implies Cardinal Inequality
- Subset of Ordinal implies Cardinal Inequality
- Sum of Cardinals is Associative
- Sum of Cardinals is Commutative
- Surjection iff Cardinal Inequality