Does the "inverse elements of addition" axiom hold for the following set?

Question

Let $\mathbf{S} = \{ (a_1, a_2): a_1, a_2 \in \mathbf{R} \}$. Where for $a = \left(a_1, a_2\right), b = \left(b_1, b_2\right) \in \mathbf{S}$ and $c \in \mathbf{R}$ $$a+b = (a_1, a_2) + (b_1, b_2) = (a_1 + b_1, 0)$$ and $$ca=(ca_1, 0)$$

In the book I am reading now it is written that $\mathbf{S}$ is not a vector space, because other axioms including the 4th axiom, i.e. "inverse element of addition", doesn't hold. But I can show that there exists such a vector. Let $a=(a_1, a_2) \in \mathbf{S}$ then $v = (v_1, v_2) \in \mathbf{S}$ where $v_1 = -a_1$ is an additive inverse of $a$: $$(a_1, a_2) + (v_1, v_2) = (a_1 + v_1, 0 ) = (a_1 - a_1, 0 ) = (0, 0)$$

Is it a typo in the book? Or I am missing something? Thanks in advance.

Answer 1

I'll turn my comment into an answer:

I suspect that the main problem is that there's no element which can act as zero: $(0,0)$ does not, in fact, act as an additive identity. And once you've accepted that, what do you mean by "additive inverse"? The only thing I've ever used it for is "thing that you add to something to get additive identity." If there's no additive identity (nothing that we can actually call $0$, no matter how much like a $0$ it looks like) then that definition doesn't even make sense.

At this point there are three ways to interpret the fourth axiom and it's truth status:

• $0$, as part of the signature of the logic, does refer to some particular element. It's just that, due to the failure of the earlier axiom, that element $0$ doesn't have the property we expect of it. Under this interpretation, axiom 4 is true but irrelevant.
• $0$ refers to the element designated by the additive identity axiom. Since that axiom failed, $0$ was never even specified and the fourth axiom isn't properly stated, never-mind true or false.
• The fourth axiom should be understood as stating that there is some element $-a$ such that $a + (-a)$ is an additive identity. Under this interpretation, axiom 4 is just false: there are no additive identities.

I lean towards the second interpretation. I guess your book prefers the third.