There has been an ongoing debate whether the correct form after “than” is I, he, she, we, and they or me, him, her, us, and them. Consider, for example, the following three sentences:
2) Peter is smarter than me.
3) Peter is smarter than I.
Is any of them wrong? No. English has no regulatory body which could give a definitive answer to what is considered unacceptable. What matters is how people use the language, and all three forms are commonly seen in professional writing.
Smarter than I am is the most common form in formal writing. The reason is that probably all English speakers consider it acceptable, while there is a certain amount of controversy surrounding the other two.
Smarter than me is the most common form in spoken language and also the second most common one in English literature, so it can hardly be considered wrong. Don’t be afraid to use it. That’s just how the language developed.
Smarter than I has been traditionally understood as a short way of saying “smarter than I am”. Some grammarians argue that “than” is always a conjunction (like “so”, “while”, “because”, etc.) and what follows must be a whole clause, where the verb may possibly be implied; hence they argue that “than me” is wrong because “than me am” doesn’t make sense, and therefore one should say “than I”.
Nonetheless, the third form is the least common one of the three when we examine the existing body of modern English literature, and it is quite uncommon in spoken language, to such a degree that a great number native speakers would even consider “than I” to be an error. Thus, my personal recommendation is to avoid it altogether.
Cases where one of the variants is wrong
We have discussed the case when the compared things or people constituted the subject of the sentence and there was no object. However, consider the following situation:
There are two ways to describe the situation:
The third option doesn’t work here because the implied meaning would be that “Peter likes Laura more than I like Laura”, which is not what the sentence is supposed to mean. Similarly, if
there are again two ways to express the thought:
The third sentence is not strictly speaking wrong; it follows the same pattern as example no. 2) at the beginning of the article. However, it is so likely to be misunderstood as “me being the object of John’s liking” that you should never use it.
In conclusion, if there is a possibility of misunderstanding, the best solution is to use a subordinate clause (“than I do”, “than he likes me”), which is always unambiguous.