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I read this while browsing. Basically, the thesis is that $\mathbb N \subset \mathbb Z$ is false, because actually $\mathbb Z_{\ge0}$ is only isomorphic and not actually equal to $\mathbb N$, the same way "complex real numbers" ($z \in \mathbb C : \exists x \in \mathbb R : z = x + 0i$) aren't actually real numbers, but only isomorphic.

So is the same valid for $\mathbb N \subset \mathbb Z$, or for $\mathbb Z \subset \mathbb Q$ (since one could argue that $\mathbb Q \subset \mathbb Z \times \mathbb Z_{>0}$ and $(2,1)$ is different than $2$, but I'm not sure), or for $\mathbb Q \subset \mathbb R$?

If yes, is there any circumstance where this subtle distinction actually matters? That is, is there a particular situation where assuming $1 + 0i = 1$ or $1 = +1$ (where $1\in\mathbb N$ and $+1\in\mathbb Z$) leads to wrong conclusions?

Thanks in advance.

Elvis
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1 Answers1

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The usual approach is: once the complex numbers have been properly defined, we interpret the symbols $\Bbb N,\Bbb Z,\Bbb R$ as denoting the appropriate subsets, making $\Bbb N\subset\Bbb Z\subset\Bbb R\subset \Bbb C$ literally true. This is obviously convenient, and there's no technical problem with it - it's just a matter of notation.

Karl
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