# Does the restriction of the well ordering principle to a fixed uncountable cardinal imply the general well ordering principle?

The Well-Ordering principle states that for every set $X$, there exists a well-ordering $\leq$ on $X$.

Let us not assume the Axiom of Choice. Let $\alpha$ be an uncountable cardinal and suppose the following is true: For every set $X$ of cardinal $\alpha$, there exists a well-ordering $\leq$ on $X$.

Can one now prove the well-ordering principle on any cardinal?

I ask this because, in a conversation with some people, i heard the claim that the existence of the Vitali set is equivalent to the axiom of choice, and therefore to the well-ordering principle. But now, i highly doubt that is true, since in the construction of the Vitali Set, one only need to assume that the well-ordering principle (or the axiom of choice, depending on how you take the crucial step) is true for sets of cardinal $2^{\aleph_0}$.

Nevertheless, if the restriction of the well-ordering principle to $2^{\aleph_0}$ imply the general well-ordering principle, i can totally see how this would be true.

The answer is negative. And you are correct. To construct a Vitali set one only need choice up to $2^{\aleph_0}$, or that the real numbers can be well ordered (which might happen regardless to how badly choice fails in the universe).