I want to prove that every partial order on a finite set $X$ can be extended to a linear order and I want to do it without assume the axiom of choice.
I try to do a proof by induction on the cardinality of the set $X$.
Base case: If $|X|=0$ then $X=\emptyset$, then the unique partial order on $X$ is given by the void relation, that is also a linear order. If $|X|=1$ then $X= \{ x \} $, then the unique partial order on $X$ is given by the relation $x \le x$, that is also a linear order.
Induction step: Suppose that every partial order on a set with cardinality $n-1$ can be extended to a linear order. Let's consider a partial order on a set $X$ with cardinality $n$. How can I extend this partial order to a total order?