# Incompleteness of Robinson's Arithmetic

In Smith's book about Godel's Theorems a proof is given of incompleteness of Robinson Arithmetic. It's done by producing a special interpretation of the theory:

1. Domain is $$N^* = N \cup \{a,b\}$$, a and b are "rogue elements"
2. Sum $$+^*$$ as follows: $$a +^* n = a, n \in N \\ b +^* n = b, n \in N \\ x +^* a = b, x \in N^* \\ x +^* b = a, x \in N^*$$
3. From this he concludes $$0 +^* a \neq a$$. How is this reached?

Thanks for any guidance.

• Notice how a tiny bit of formatting transforms your post from barely readable into very clear. You can click the edit timestamp to see how it is done. You can also look into the FAQ on formatting. – rschwieb Feb 13 at 17:29
• @MauroALLEGRANZA I see the accepted answer there appears to be the problem statement here. So rather than a duplicate, I think the user apparently has a question about a step in the answer. – rschwieb Feb 13 at 17:31
• Doesn't the assumption that $a$ and $b$ are distinct, and the third of the four items say $0+^* a=b\neq a$? – rschwieb Feb 13 at 17:33
• I don't see anywhere in the linked duplicate that actually answers the users' question, but I'm not going to get any heartburn about it because I think it might be already covered by my comment above. – rschwieb Feb 13 at 17:36
• I'll try to do better with the formatting next time. Thanks for your help. Agapito – Agapito Martinez Feb 13 at 18:58