What does these axioms in Robinson arithmetic mean?

Question

I am trying to understand the axioms of Robin arithmetic, but am puzzled by these axioms: "

4)x + 0 = x

5)x + y′ = (x + y)′

6)x × 0 = 0

7)x × y′ = (x × y) + x

The intended interpretation of “ x′ ” is the successor function, and obviously, of + and ×, the addition and the multiplication functions, respectively. “0” is the only constant and denotes the number zero. "

I understand 4 and 6, but am puzzled by 5 and 7. From what I've read 4)5) and 6)7) are used to define addition and multiplication, respectively.

Answer 1

5 means "$x+(y+1)=(x+y)+1$," and 7 means "$x\cdot (y+1)=x\cdot y+x$." Basically, these tell you how to compute a sum/product of "big" numbers by repeatedly breaking it down into sums/products of "small" numbers.

For example, according to the rules, we can compute $0''+0''$ as follows:

• $0''+0''=(0''+0')'$

• $(0''+0')'=([0''+0]')'$

• $([0''+0]')'=0''''$

Hooray! We've shown $2+2=4$!

As a "fun" exercise, try using Robinson's axioms to show that $0''\times 0'''=0''''''$ ($2\times 3=6$).

Answer 2

In ordinary arithmetic, the "successor" function is just adding 1. So (5) becomes $$x+(y+1)=(x+y)+1$$ and (7) becomes $$x\times(y+1)=(x\times y)+x.$$ These are hopefully familiar true statements from ordinary arithmetic (the first is an instance of the associative property of addition, and the second is an instance of the distributive property of multiplication over addition together with the fact that $x\times 1=x$). So these axioms are saying that the operations $'$, $+$, and $\times$ have certain properties that they ought to have if they are supposed to behave like ordinary arithmetic.