TITLE: Ordinals and complexity classes
QUESTION [10 upvotes]: What is the least recursive ordinal $\alpha$ such that there is no algorithm in complexity class $\mathsf{P}$ which implements a well-ordering of $\mathbb{N}$ with order type $\alpha$? (where the size of input is the total number of digits in numbers being compared) 
Is it true that no well-ordering of $\mathbb{N}$ with order type $>\alpha$ can be implemented using an algorithm in $\mathsf{P}$?
Has connection between ordinals and complexity classes been studied? Can you recommend any books or papers related to this topic?

REPLY [3 votes]: You may want to check "Dynamic Ordinal Analysis" by Arnold Beckmann which is an attempt to define a finer notion to classical ordinals that can be used ti distinguish between complexity classes.