TITLE: What about the classification of big finite simple groups?
QUESTION [10 upvotes]: How hard is it to classify all big finite simple groups, i.e., all finite simple groups larger than some sufficiently large constant? Alternatively - how hard is it to classify all finite simple groups up to finitely many exceptions?
Is any known proof of such a classification significantly easier or shorter than the classification of (all) finite simple groups?

REPLY [2 votes]: Hrushovski asks roughly this question in his paper On Pseudo-Finite Dimensions.  In section 6 he discusses the "classification of large finite simple groups" (in analogy to some of the related results which can be proven by model theoretic means, and end up only giving information about sufficiently large finite objects), and notes that the only known proof is via the full classification of finite simple groups.