Natural Numbers - kipawaa/Proof-Tree GitHub Wiki

Statement

Common Definition

The natural numbers are defined to be the set $\mathbb{N} = \{0, 1, 2, 3, \dots\}$.

Set-Theoretic Definition

The natural numbers are the elements of the set $\mathbb{N} = \{x \in A \mid x \in I$ for every inductive set $I\}$.

Explanation

This set exists due to the Axiom of Infinity.

Proof(s)

History

There is debate as to whether 0 is included in this set. Some argue that the natural numbers should reflect reality, where you cannot have 0 of something. I argue that I have 0 Ferraris in the real world.

Alternatively, it is argued that 0 should be included to provide a convenient distinction from the set $\mathbb{Z}^+ = \{1, 2, 3, \dots\}$.

Applications

Links

Dependencies

• Axiom of Infinity

Dependents

• Integers
• Cardinal Numbers
• Countable
• Finite
• Usual Ordering of the Natural Numbers
• Natural Numbers Are Transitive
• Omega Ordinal
• Sequence

Sources

• Hrbacek, K., & Jech, T. (1999). Introduction to Set Theory, Revised and Expanded (3rd ed.). CRC Press. https://doi.org/10.1201/9781315274096