Existence of total order for every set

Question

please prove it from Compactness theorem for propositional logic. Don't assume AC in any form. I mean relation $<$ is total order for $X$ iff

1. trichotomy
2. transitivity
3. irreflexivity

are true about $<$ on $X$

Answer 1

Consider the language that has propositional variables $p_{x,y}$ for any pair $x,y\in X$. It will effectively stand for $x<y$.

Now, our theory $T$ will be the combination of these three theories:

1. $\lnot p_{x,x}$ for all $x\in X$;
2. $p_{x,y}\lor p_{y,x}$ for all $x\neq y$; and
3. $p_{x,y}\land p_{y,z}\rightarrow p_{x,z}$ for all $x,y,z\in X$.

Use compactness to prove that $T$ is satisfiable: for any finitely many propositions, only finitely many variables are involved, and we can linearly order the set of necessary $x$'s to find a satisfying assignment for the variables.

So by compactness there is an assignment $\sigma$ for which $T$ is evaluated as true. Now define $x<y$ if and only if $\sigma(p_{x,y})=\rm True$, and prove that $<$ is a linear order.