Definition:Natural Numbers/Inductive Sets in Real Numbers

Definition

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

Let $\II$ be the set of all inductive sets defined as subsets of $\R$.

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

$\N := \ds \bigcap \II$

where $\ds \bigcap$ 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