# Category:Natural Numbers

This category contains results about the structure of the set of natural numbers $\N$.

Definitions specific to this category can be found in Definitions/Natural Numbers.

The **natural numbers** are the counting numbers.

The **set of natural numbers** is denoted $\N$:

- $\N = \set {0, 1, 2, 3, \ldots}$

## Subcategories

This category has the following 13 subcategories, out of 13 total.

### M

### N

### P

### T

## Pages in category "Natural Numbers"

The following 85 pages are in this category, out of 85 total.

### A

### C

### I

- Identity Element of Natural Number Addition is Zero
- Identity Element of Natural Number Multiplication is One
- Inclusion of Natural Numbers in Integers is Epimorphism
- Infinite Set of Natural Numbers is Countably Infinite
- Initial Segment of Natural Numbers determined by Zero is Empty
- Initial Segment of One-Based Natural Numbers determined by Zero is Empty
- Inverse Completion of Natural Numbers
- Invertible Elements under Natural Number Multiplication

### M

### N

- Natural Number Addition Commutativity with Successor
- Natural Number Addition Commutes with Zero
- Natural Number Addition is Associative
- Natural Number Addition is Cancellable
- Natural Number Addition is Cancellable for Ordering
- Natural Number Addition is Closed
- Natural Number Addition is Commutative
- Natural Number Multiplication Distributes over Addition
- Natural Number Multiplication is Associative
- Natural Number Multiplication is Cancellable
- Natural Number Multiplication is Cancellable for Ordering
- Natural Number Multiplication is Closed
- Natural Number Multiplication is Commutative
- Natural Number Ordering is Transitive
- Natural Number Subtraction is not Closed
- Natural Numbers are Comparable
- Natural Numbers are Infinite
- Natural Numbers are Non-Negative Integers
- Natural Numbers as Cardinals
- Natural Numbers Bounded Below under Addition form Commutative Semigroup
- Natural Numbers form Commutative Semiring
- Natural Numbers form Inductive Set
- Natural Numbers form Subsemiring of Integers
- Natural Numbers have No Proper Zero Divisors
- Natural Numbers Set Equivalent to Ideals of Integers
- Natural Numbers under Addition do not form Group
- Natural Numbers under Addition form Commutative Monoid
- Natural Numbers under Addition form Commutative Semigroup
- Natural Numbers under Multiplication do not form Group
- Natural Numbers under Multiplication form Ordered Commutative Semigroup
- Natural Numbers under Multiplication form Semigroup
- Natural Numbers under Multiplication form Subsemigroup of Integers
- Non-Empty Subset of Initial Segment of Natural Numbers has Greatest Element
- Non-Zero Natural Numbers under Addition do not form Monoid
- Non-Zero Natural Numbers under Addition form Semigroup
- Non-Zero Natural Numbers under Multiplication form Commutative Monoid
- Non-Zero Natural Numbers under Multiplication form Commutative Semigroup
- Not every Non-Empty Subset of Natural Numbers has Greatest Element
- Nth Power of a Natural Number Introduces no New Prime Factors

### O

- Order Topology on Natural Numbers is Discrete Topology
- Ordering on Natural Numbers is Compatible with Addition
- Ordering on Natural Numbers is Compatible with Addition/Corollary
- Ordering on Natural Numbers is Compatible with Multiplication
- Ordering on Natural Numbers is Compatible with Multiplication/Corollary
- Ordering on Natural Numbers is Trichotomy

### P

### S

- Second Principle of Recursive Definition
- Set of Natural Numbers is Primitive Recursive
- Strictly Increasing Infinite Sequence of Positive Integers is Cofinal in Natural Numbers
- Subset of Natural Numbers is Cofinal iff Infinite
- Subset of Natural Numbers under Max Operation is Monoid
- Surjection from Natural Numbers iff Countable
- Surjection from Natural Numbers iff Right Inverse