I need to prove the following:
"Let $ \mathbb{N}$, if $a \in \mathbb{N} \to a+1 \in \mathbb{N}$"
thanks in advance!!
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6Well, what is your definition of $\mathbb{N}$? – user79202 May 25 '13 at 19:51
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2Also define $+$, please. – Git Gud May 25 '13 at 19:52
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2@Garnak Olegovitc this could be a part of the axiomatic definition of natural numbers – Federica Maggioni May 25 '13 at 19:52
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2@GarnakOlegovitc Well, you have nothing to prove then, simply accept that – Federica Maggioni May 25 '13 at 19:55
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There is actually something to prove here. $a+1$ is defined in the Peano axioms as $a+S(0)=S(a+0)=S(a)$. Hence, if $a\in\mathbb{N}$, $a+1=S(a)\in \mathbb{N}$.
We can go further; $a+2=a+S(1)=S(a+1)=S(S(a))$. Hence, if $a\in \mathbb{N}$, $a+2=S(S(a))\in \mathbb{N}$. In fact, for any specific $k$, we can equally prove that if $a\in \mathbb{N}$ then $a+k\in \mathbb{N}$.
vadim123
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