# Category:Natural Numbers

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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 18 subcategories, out of 18 total.

### M

### N

### O

### P

### T

## Pages in category "Natural Numbers"

The following 102 pages are in this category, out of 102 total.

### C

- User talk:Calogero
- Composition of Addition Mappings on Natural Numbers
- Consecutive Subsets of N
- Continuum Property implies Well-Ordering Principle
- Convergent Series of Natural Numbers
- Cross-Relation Equivalence Classes on Natural Numbers are Cancellable for Addition
- Cross-Relation on Natural Numbers is Equivalence Relation

### E

### 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
- Inductive Construction of Natural Numbers fulfils Peano's Axiom of Injectivity
- Inductive Construction of Natural Numbers fulfils Peano's Axioms
- 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

### 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 has Same Prime Factors as Integer Power
- Natural Number is not Subset of its Union
- Natural Number is Ordinary Set
- Natural Number is Superset of its Union
- Natural Number is Transitive Set
- Natural Number is Union of its Successor
- 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 Commutative
- Natural Number Ordering is Transitive
- Natural Number Subtraction is not Closed
- 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 cannot be Elements of Each Other
- 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 do not form Group/Corollary
- Natural Numbers under Addition form Commutative Monoid
- Natural Numbers under Addition form Commutative Semigroup
- Natural Numbers under Multiplication form Ordered Commutative Semigroup
- Natural Numbers under Multiplication form Semigroup
- Non-Empty Set of Natural Numbers with no Greatest Element is Denumerable
- 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
- Nonzero natural number is another natural number successor
- Not every Non-Empty Subset of Natural Numbers has Greatest Element

### 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

- Positive Rational Numbers under Addition not Isomorphic to Natural Numbers
- Power Set of Natural Numbers is Cardinality of Continuum
- Power Set of Natural Numbers is not Countable
- Principle of Counting
- Principle of Recursive Definition
- Principle of Recursive Definition/General Result
- Properties of Natural Numbers

### S

- Second Principle of Recursive Definition
- Set of Natural Numbers can be Derived using Comprehension Principle
- Set of Natural Numbers Equals its Union
- Set of Natural Numbers Equals Union of its Successor
- Set of Natural Numbers is either Finite or Denumerable
- Set of Natural Numbers is Primitive Recursive
- Strictly Increasing Infinite Sequence of Positive Integers is Cofinal in Natural Numbers
- Strictly Positive Integers have same Cardinality as Natural Numbers
- Subset of Natural Numbers is Cofinal iff Infinite
- Subset of Natural Numbers under Max Operation is Monoid
- Subset of Naturals is Finite iff Bounded
- Successor Mapping on Natural Numbers has no Fixed Element
- Surjection from Natural Numbers iff Countable
- Surjection from Natural Numbers iff Right Inverse