Can $\mathbb{N}$ be an element of $\mathbb{R}$

Question

Is $\mathbb{N}$ a element of $\mathbb{R}$ I understand that $\mathbb{N}$ ⊆ $\mathbb{R}$, but does that also imply that $\mathbb{N}$ ∈ $\mathbb{R}$?

Answer 1

[I will answer this question naively, ignoring the actual set theoretical constructions of either sets in question.]

No, elements of $\mathbb{R}$ are numbers, no sets of numbers. So it does not make sense to say that $\mathbb{N}\in \mathbb{R}$. But it does make sense to say that $\mathbb{N}\subseteq \mathbb{R}$ because every element of the first set (every natural nubmer) is also an element of the second set (is also a real number).