Show that given a partial order there exists a total order

Question

STATEMENT: Suppose that $≺$ is a partial ordering of $\mathbb{N}$. Use the Compactness Theorem for first order logic to show that there is a total ordering $≺_∗$ of $\mathbb{N}$ such that for all n and m in $\mathbb{N}$,if $n≺m$ then $n≺_∗ m.$

QUESTION: I am unsure of how to proceed with this type of problem. My guess was to create a set of sentences such that $(\forall m,n\in \mathbb{N}(n≺m \rightarrow n≺_∗ m)$ union with another set of sentences $(\forall m,n\in \mathbb{N}((m\not≺ n) \wedge (\forall k\in\mathbb{N}(k≺m\rightarrow k≺n))\rightarrow m≺_*n).$ Then showing that each finite subset is satisfiable means that our set of sentences has a model $(\mathbb{N},≺_*)$.

Using Zorn's lemma you can show that every partial ordering can be extended to total ordering, no matter the set. — brick, Dec 04 '14 at 19:32
I know how to prove it using Zorn's lemma, but I don't know how to prove it using compactness theorem. — Enigma, Dec 04 '14 at 19:34

score 1 · Accepted Answer · answered Dec 04 '14 at 19:48

Your idea is fine, however, I would use a differed set of sentences, such as (for all $n,m,k\in\mathbb N$)

$n\prec m\to n\prec_*m$
$\neg (n\prec_* n)$
$n\prec_*m\lor m\prec_*n\lor n=m$
$n\prec_*m\land m\prec_*k\;\to\; n\prec_*k$

which jointly precisely state that $\prec_*$ extends $\prec$, is irreflexive, total, and transitive.

Given only a finite set of these sentences, they involve only finitely many numbers. On these, we can perform a topological sort according to $\prec$ and use the result to defined a total order $\prec_*$ extending $\prec$ on this finite set.

If topological sort sounds like too much borrowing from the Computer Science department, you may want to prove

Lemma. If $X$ is a nonempty finite set partially ordered by $\prec$, then there exists $x\in X$ such that $\forall y\in X\colon x\not\prec y$.

and use this to show by induction on the size of the set that every partial order on a finite set can be extended to a total order.

Show that given a partial order there exists a total order

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