# 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 numbers used for counting.

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

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

## Subcategories

This category has the following 15 subcategories, out of 15 total.

### E

- Extended Natural Numbers (empty)

### M

### N

- Natural Number is Ordinal (3 P)
- Natural Numbers/1-Based (20 P)

### O

### P

### T

- Transfinite Arithmetic (4 P)

## Pages in category "Natural Numbers"

The following 84 pages are in this category, out of 84 total.

### B

### C

- Cardinality of Power Set of Natural Numbers Equals Cardinality of Real Numbers
- Condition for Equivalence Relation for Max Operation on Natural Numbers to be Congruence
- Consecutive Subsets of N
- Continuum Property implies Well-Ordering Principle
- Convergent Series of Natural Numbers
- Cross-Relation on Natural Numbers is Equivalence Relation

### E

- Element of Natural Number is Natural Number
- Equality of Natural Numbers
- Equality of Terms of Natural Numbers is Provable
- Equivalence Classes of Cross-Relation on Natural Numbers
- Existence of One-Sided Inverses on Natural Numbers whose Composition is Identity Mapping
- Existential Quantification of Provable Arithmetic Formula is Provable

### I

- 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
- Inequality of Natural Numbers is Provable
- 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

### N

- Natural Number has Same Prime Factors as Integer Power
- Natural Number is not Subset of its Union
- Natural Number is Ordinal
- 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 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 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 Min Operation forms Total Semilattice
- Natural Numbers with Divisor Operation is Isomorphic to Subgroups of Integer Multiples under Inclusion
- Natural Numbers with Divisor Operation is Isomorphic to Subgroups of Integer Multiples under Inclusion/Corollary
- Non-Empty Set of Natural Numbers with no Greatest Element is Denumerable
- Non-Empty Subset of Initial Segment of Natural Numbers has Greatest Element
- Nonzero Natural Number is Successor
- Not every Non-Empty Subset of Natural Numbers has Greatest Element

### O

### P

### S

- Second Principle of Recursive Definition
- Set of Natural Numbers can be Derived using Axiom of Abstraction
- Set of Natural Numbers Equals its Union
- Set of Natural Numbers Equals Union of its Successor
- Set of Natural Numbers is Limit Ordinal
- Set of Natural Numbers is Ordinal
- Set of Natural Numbers is Primitive Recursive
- Set of Natural Numbers is Smallest Ordinal Greater than All Natural Numbers
- 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 is either Finite or Denumerable
- 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