If I am not wrong, the well-ordering theorem is strictly stronger than the axiom of choice in second-order logic. I am not sure to understand how this is possible. The reason is that second order logic is supposedly more restrictive than first order logic, because it is categorical. Thus, any theory in first order logic should have more models (many of them unintended, or non-standard) than the same theory in second order logic. In first order logic, within ZF, the axiom of choice is equivalent to the well-ordering principle. I interpret this as meaning that there are no models in which the axiom of choice is true but the well-ordering principle is false. However it seems that in second order logic we can have models in which the axiom of choice is true but the well-ordering principle is false. Am I correct?
UPDATE: this is possibly a duplicate question, as mentioned by Andres. But just to be sure that I correctly understood the "duplicate answer": Would the conclusion be that the statement "there are second-order models in which the axiom of choice is true but the well-ordering principle is false but there are no first-order models in which the axiom of choice is true but the well-ordering principle is false" is false?