Field axioms - is there an axiom for multiplication with zero?

[musicgold]

Homework Statement

This is not a homework problem.
I am reading a proof and not sure why a particular step is taken / not taken.

Relevant Equations

why is there no axiom like X . 0 = 0

Please refer to the screenshot below. Every step is justified with an axiom. Please see the link to the origal document at the bottom.

I am trying to understand why the proof was not stopped at the encircled step.
1. Is there no axiom that says $x \cdot 0 = 0$ ?
2. Isn't the sixth step using the fact that a 0 can be represented by $x \cdot 0$ ?

Document

 

[mfb]

The line you marked is just using 1+(-1)=0, replacing the left hand side by the right hand side in the previous line.

 

[Dr.AbeNikIanEdL]

The sixth step is using the fact that $a + (-a)=0$, with $a = x\cdot0$. Afaik, $0$ only appears in the axioms when defining addition, that it has any special role for multiplication needs to be derived.

 

[member 587159]

$x.0=x.(0+0)= x.0+x.0$
Add $-x.0$ to both sides to obtain $x.0=0$.

 

[hilbert2]

I have sometime wondered whether it would be possible to state the axioms of $\mathbb{R}$ in a different way so that $0\cdot x=0$ would be one of them but they would still describe exactly the same number system. The normal set of axioms doesn't contain that property, it has to be derived.

You can't start listing every possible truth about the real number system, anyway, because there's an infinite number of those truths. Actually, it's too infinite to even form a proper set (in the sense of set theory), as the Patrick Grim's proof for "There is no set of all truths" should also apply to any statement "There is no set of all truths about entity X".

http://www.pgrim.org/articles/grim_no_set_of_all_truths.pdf

 

[HallsofIvy]

There is no "axiom" that x.0= 0 because it can be proven from the other axioms. By "distribution", x(y+ 0)= xy+ x0. But y+ 0= y so x(y+ 0)= xy. From xy= xy+ x0, add the "additive inverse" of xy to both sides to get x0= 0.

 

[WWGD]

I have sometime wondered whether it would be possible to state the axioms of $\mathbb{R}$ in a different way so that $0\cdot x=0$ would be one of them but they would still describe exactly the same number system. The normal set of axioms doesn't contain that property, it has to be derived.

You can't start listing every possible truth about the real number system, anyway, because there's an infinite number of those truths. Actually, it's too infinite to even form a proper set (in the sense of set theory), as the Patrick Grim's proof for "There is no set of all truths" should also apply to any statement "There is no set of all truths about entity X".

http://www.pgrim.org/articles/grim_no_set_of_all_truths.pdf

In my experience this is called the full theory,aka, all statements /wffs that can be assigned a truth value.