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?

$\mathbb{R}$ contains all natural numbers but $\mathbb{R} \neq \mathbb{N}$. — Giorgos Giapitzakis, Dec 07 '23 at 13:46
@GiorgosGiapitzakis's example was the one that came to my mind when I started to think of examples where the larger set had cardinality greater than $\aleph_0$. — Henrik supports the community, Dec 07 '23 at 14:09

QuantumSuperfield · Answer 1 · 2023-12-07T14:18:26.603

2

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.

edited Dec 07 '23 at 14:18

answered Dec 07 '23 at 14:10

QuantumSuperfield

2,661

3

I would use $\subseteq$ rather than $\subset$ here, because the latter one usually means a proper subset – Smylic Dec 07 '23 at 14:16
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Thank you for your comment. – QuantumSuperfield Dec 07 '23 at 14:19

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

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