Is a set $T$ containing all natural numbers identical to $\mathbb{N}$?

Question

In this comment, I see that Mauro Allegranza and I were saying the same thing as shown in the table below:

Allegranza	Zeynel
$0 \in T$	$0 \in T$
$n \in T$	$n \in T$
$S(n) \in T$	$S(n) \in T$
$ T = \mathbb{N}$	$T$ contains all natural numbers

Allegranza claims my version is not correct. But to me "$ T = \mathbb{N}$" and "$T$ contains all natural numbers" are identical statements. Is there a difference I don't understand?

Answer 1

But to me "$T=\Bbb{N}$" and "$T$ contains all natural numbers" are identical statements.

$T=\Bbb{N}$ means that $T\subseteq\Bbb{N}$ and $\Bbb{N}\subseteq T$. On the other hand, the statement "$T$ contains all natural numbers" only means that $\Bbb{N}\subseteq T$.

Therefore, "$T=\Bbb{N}$" and "$T$ contains all natural numbers" are different statements.