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I am learning proofs with $\mathbb N $. I don't have significant problems using the axioms to prove propositions, I have a problem understanding certain axioms and the definition of >.

1) If $m,n \in\mathbb N$, then $m + n \in\mathbb N$

2) If $m,n \in\mathbb N$, then $mn \in\mathbb N$

3) $0 \notin\mathbb N$

4) For every $m \in\mathbb Z$, we have $m \in\mathbb N$ or $m = 0$ or $-m \in\mathbb N$

I don't understand why 3) is "obvious" since we're defining a new subset ($\mathbb N$). I mean, why couldn't we assume $0 \in\mathbb N$?

Moreover, > is then defined as: Let $m,n \in\mathbb Z$, $m > n$ means $m - n \in\mathbb N$. Again, I am confused. Why couldn't we say m > n means $m - n \notin\mathbb N$? I am sorry if the questions are very "basic" but since I am learning proofs, I am introduced concepts progressively, and I am confused. I am also sorry if the question tag is not appropriate. Please let me know which one is better. Thank you!

Hanul Jeon
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Johnathan
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  • I think your confusion might be that these are definitions, meaning to capture some intuition (i.e. defining $m>n$ to be $m-n\in \mathbb N$ captures "$m$ is bigger than $n$ if there is some amount we can increase $n$ by to get to $m$") - but they'll certainly not come from proof. That we should not include $0$ in $\mathbb N$ follows if we define $>$ that way - as otherwise we get $1>1$ because $1-1=0\in \mathbb N$. Of course, for other purposes you could (and mathematicians so often do) consider $0\in \mathbb N$. – Milo Brandt Mar 02 '15 at 04:23

1 Answers1

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$\mathbb{N}$ is defined as the set of natural numbers, that is $$\mathbb{N}=\{1,2,3,4,5,6,7,\ldots\}.$$ You might notice that $0$ is not included in the above enumeration. So $0 \not\in \mathbb{N}$.

As for your second question, if $m,n \in \mathbb{Z}$ (if $m,n$ are whole numbers, or integers), then $m-n \in \mathbb{Z}$. Also, the given statement $m>n$ implies and $m-n > 0$. Putting those two facts together, we see that $m-n$ is positive and is an integer. So $m-n$ is a natural number, that is, $m-n \in \mathbb{N}$.

The only time $m-n \not\in \mathbb{N}$ is if we are given $m \le n$. Namely, if $m = n$, then $m-n = 0 \not\in \mathbb{N}$; similarly, if $m < n$, then $m-n < 0 \not\in \mathbb{N}$. But for what it's worth, $m-n \in \mathbb{Z}$ still, though just a nonpositive integer (a negative integer or $0$).

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