# Definition:Natural Numbers/Natural Numbers in Real Numbers

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## Definition

Let $\R$ be the set of real numbers.

Let $\mathcal I$ be the set of all inductive sets in $\R$.

Then the **natural numbers** $\N$ are defined as:

- $\N := \displaystyle \bigcap \mathcal I$

where $\displaystyle \bigcap$ denotes intersection.

It follows from the definition of inductive set that according to this definition, $0 \notin \N$.

## Also see

- Results about
**the natural numbers in $\R$**can be found here.

## Sources

- 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers