I am trying to rigorously understand how algebra works by deriving everything from the axioms of the real numbers. I thought this wouldn't be too difficult but it seems I don't have any idea where to start.
I assume the field axioms of the real numbers, plus the following axioms to establish total order:
- Antisymmetry: $(a \geq b ~\wedge~b \geq a) \rightarrow a = b$
- Transitivity: $(a \geq b ~\wedge~ b \geq c) \rightarrow a \geq c$
- Totality: $a \geq b \vee b \geq a$
and also the following:
- $a \geq b \rightarrow a + c \geq b + c$
- $a \geq 0 \wedge b \geq 0 \rightarrow ab \geq 0$
Other than the completeness axiom, I assume that these are all of the axioms I need to prove any of the properties of the real numbers (Note: these all come from my understanding of the Wikipedia article, so if it is wrong please let me know).
Having this, I now want to prove the following basic identities:
- $ a = b \rightarrow a + c = b + c$
- $ a = b \rightarrow ac = bc$
- $ a \geq b \wedge c \geq 0\rightarrow ac \geq bc$
- $a \geq b \wedge b \lt 0 \rightarrow ac \leq bc$
To be honest I have no idea where to start with this. I started working on the first by trying to prove that $ a = b \rightarrow (a \geq b ~\wedge~b \geq a)$ by somehow combining the antisymmetry and totality axiom, but failed miserably. I thought that if I could do this, it would be an easy step to show 6. from 4. But now that I think of it I really don't know how to do that either. Any help at all, even a pointer in the right direction would be a start!
1 and 4 give you 6 immediately.
7 is derived from the axioms of the product, nothing to do with order here I'd say.
8 follows from 4 and 5 ($a-b \ge 0$).
9 is very similar to the reciprocal of 5.