Non-Forking Types have Non-Forking Completions
Let $T$ be a complete $\mathcal L$-theory.
Let $\mathfrak C$ be a monster model for $T$.
Let $A\subseteq B$ be subsets of the universe of $\mathfrak C$.
Let $\pi(\bar x)$ be an $n$-type over $B$.
Suppose $\pi$ does not fork over $A$.
We will use Zorn's Lemma to find a candidate for the needed complete type.
Consider the collection $\Pi$ of all non-forking sets $\pi'$ of $\mathcal L$-formulas with parameters from $B$ such that $\pi'$ contains $\pi$.
Order $\Pi$ by subset inclusion.
Thus, by Zorn's Lemma, there is a maximal (with respect to subset inclusion) $p$ in $\Pi$.
Suppose $p$ is not complete.
- By definition, for some $\phi(\bar x, \bar b)$, $p$ contains neither $\phi(\bar x, \bar b)$ nor $\neg\phi(\bar x, \bar b)$.
- Since $p$ is non-forking, by Formula and its Negation Cannot Both Cause Forking, at least one of $p\cup\phi(\bar x, \bar b)$ or $p\cup\neg\phi(\bar x, \bar b)$ is non-forking as well.
- Hence $p$ is not maximal in $\Pi$, contradicting the choice of $p$.
Thus $p$ is complete.
Axiom of Choice
Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.
However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.