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I am solving some problems on order isomorphism.

I found this old problem Prove that $(\mathbb{N},\le)$ and $(\mathbb{Z}, \le)$ are not order isomorphic

I has solved this problem as follows:

Let $f: \mathbb{Z} \rightarrow \mathbb{N}$ is an order isomorphism. So, since $1\in \mathbb{N}$, $1=f(a)$ for $a\in \mathbb{Z}$. Now $a-1<a$ implies $f(a-1)<f(a) = 1$. So, $f(a-1) \in \mathbb{N}$ with $f(a-1)<1$. Hence contradiction!

Another solution: $f: \mathbb{N} \rightarrow \mathbb{Z}$ is an order isomorphism. Let $f(1) =a \in \mathbb{Z}$. Then $a-1 <a$ implies $f^{-1}(a-1)<f^{-1}(a)$. So, since $f^{-1}(a-1)>1$ we have $1<f^{-1}(a)$. So, $a= f(1)< f(f^{-1}(a)) = a$. Contradiction!

Am I correct?

user1234
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    I'd say you are. The main thing is $1$ is the least element of $\mathbb N$ and if an order isomorphism existed $1$ must either map to or be mapped from an element of $\mathbb Z$ that is the least element of $\mathbb Z$, but there is no such element..... You might want to point out why $f(a-1) < 1$ is a contradiction. – fleablood Jul 07 '22 at 16:33
  • You probably want to explain in the first example why $f(a-1) < 1; f(a-1)\in \mathbb N$ is a contradiction. And in the second you really need to explain why $f(a)=1;f:\mathbb Z \to \mathbb N$ implies $f^{-1}(a-1)> 1$ which you state without giving any reason.

    But....Um.... but what are you asking that is not asked in that question? Why should I not consider this to be a duplicate?

     – fleablood Jul 07 '22 at 16:38
  • An alternative second argument that may be more pertinent. Since $a-1 < a$ we must have $f(a-1) < f(a) =1; f(a-1)\in \mathbb N$. A contradiction. – fleablood Jul 07 '22 at 16:45
  • @fleablood in the first method $f(a-1) < 1$ is contradiction because a natural number is always greater than or equal to $1$. Am I correct now? – user1234 Jul 07 '22 at 16:59
  • Yes... I just meant you should include that explanation withing the proof. ... Of course a lot of what makes a good or poor proof is context and the expectations one has of the reader. – fleablood Jul 07 '22 at 17:24

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