The axiom of global choice. Technically this isn't really an axiom: global choice (GC) states that there is a formula $\phi(x, y)$ such that the relation $$A\le_\phi B:= V\models \phi(A, B)$$ is a well-ordering of the universe $V$. This can't be stated as a single formula, so in some sense it's a meta-axiom. It clearly implies choice, and is implied by $V=L$: we already have a partition of the universe into $L_0$, $L_1$, . . . , $L_\alpha$, . . ., and we can get from here to a full (class-)well-ordering of the universe by fixing at the outset some well-ordering of formulas in the language of set theory (since at each stage in the construction of $L$ we are only taking definable powersets; this is why this argument doesn't work in just $ZFC$).
• Although the relative consistency of Choice was proven rather early, it was via $L$, which satisfies $GC$ as well; the result that, assuming the consistency of $ZFC$, there is a model of $ZFC$ with no definable well-ordering of the universe came much later [NOTE: this is based on foggy memory, and I don't recall exactly when this result happened; can someone remind me?], and generally telling whether a model of $ZFC$ satisfies $GC$ is very hard.
• Basically by reversing the argument that it is true in $L$, one can make an informal argument that GC implies that the universe is small (contains only "buildable" things). So there's a somewhat intuitive argument for $AC+\neg GC$: the universe should be "big enough" that for each family of sets, we have a choice function, but the universe should also be big enough that any specific definable well-order "misses something."
• $GC$ is useful in other set theories. First of all, as noted above, even stating $GC$ needs a class theory like Morse-Kelley or NBG (expansions of $ZFC$ to also talk about classes). We can also ask about the status of $GC$ in really odd set theories like New Foundations (and, in factEDIT: As Ali points out below, we can phrase $GC$ directly in $NF$, since NF/NFU Choice and Global Choice are essentially one and the latter has a universal same - but the point still stands that global choice could still be interesting in set ; does anyone know whether theories different than $NFU+GC$ is consistent?).ZFC$.) •$GC$makes sense even outside of set theories! It doesn't make sense to ask whether a given ring$R$satisfies the axiom of choice, since elements of$R$aren't (at least on the face of things) sets, but it does make sense to ask whether there is a formula in the language of rings which well-orders$R$. 1 The axiom of global choice. Technically this isn't really an axiom: global choice (GC) states that there is a formula$\phi(x, y)$such that the relation $$A\le_\phi B:= V\models \phi(A, B)$$ is a well-ordering of the universe$V$. This can't be stated as a single formula, so in some sense it's a meta-axiom. It clearly implies choice, and is implied by$V=L$: we already have a partition of the universe into$L_0$,$L_1$, . . . ,$L_\alpha$, . . ., and we can get from here to a full (class-)well-ordering of the universe by fixing at the outset some well-ordering of formulas in the language of set theory (since at each stage in the construction of$L$we are only taking definable powersets; this is why this argument doesn't work in just$ZFC$). A couple comments on why I think global choice is interesting (even though it's not expressible in the language of set theory): • Although the relative consistency of Choice was proven rather early, it was via$L$, which satisfies$GC$as well; the result that, assuming the consistency of$ZFC$, there is a model of$ZFC$with no definable well-ordering of the universe came much later [NOTE: this is based on foggy memory, and I don't recall exactly when this result happened; can someone remind me?], and generally telling whether a model of$ZFC$satisfies$GC$is very hard. • Basically by reversing the argument that it is true in$L$, one can make an informal argument that GC implies that the universe is small (contains only "buildable" things). So there's a somewhat intuitive argument for$AC+\neg GC$: the universe should be "big enough" that for each family of sets, we have a choice function, but the universe should also be big enough that any specific definable well-order "misses something." •$GC$is useful in other set theories. First of all, as noted above, even stating$GC$needs a class theory like Morse-Kelley or NBG (expansions of$ZFC$to also talk about classes). We can also ask about the status of$GC$in really odd set theories like New Foundations (and, in fact, we can phrase$GC$directly in$NF$, since the latter has a universal set; does anyone know whether$NFU+GC$is consistent?). •$GC$makes sense even outside of set theories! It doesn't make sense to ask whether a given ring$R$satisfies the axiom of choice, since elements of$R$aren't (at least on the face of things) sets, but it does make sense to ask whether there is a formula in the language of rings which well-orders$R\$.