Prove: For any $x,y \in \mathbb{N}, y \neq x+y$.

I'm only suppose to use the Peano axioms as defined here http://aleph0.clarku.edu/~djoyce/numbers/peano.pdf and the properties of addition in $\mathbb{N}$.

I've already proved most of the properties of addition excepting this one and I don't know how to approach to this.

  • $\begingroup$ Prove by induction. $\endgroup$ – Berci Aug 24 '14 at 22:39
  • $\begingroup$ @user143201 Do you need further help with this? $\endgroup$ – Git Gud Aug 26 '14 at 10:06
  • $\begingroup$ I did solved it with your hint. But I don't know if I have to click the answer button, because what you gave me was only a hint $\endgroup$ – Keith Aug 27 '14 at 13:55
  • $\begingroup$ @user143201 It is acceptable to not do anything, to accept my answer even though is was only a hint and also to just type your own answer and accept it. $\endgroup$ – Git Gud Sep 30 '14 at 21:22

Hints: Take an arbitrary $x\in \mathbb N$ and prove the statement $\forall y\in \mathbb N(x\neq x+y)$ by induction. You'll have to use the definition of $+$, the injectivity of the successor function and the fact that $1$ is not the successor of any number.


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