You would, though it'd sound out loud as "targets's". Wow I never thought being a grammar nazi would have a real world use, turns out the paradox forum is occasionally in need.
Technically, my dear grammar nazi, there is an error in your post here as well: Paradox is a proper noun in this context and should be capitalized (which is probably on par with pointing out a missing apostrophe). One could also make the case that there should be a comma after "Wow," as it is a matter of convention to place one after non-essential introductory clauses.
I also feel like pointing out English is one of those few languages without a formal regulatory body to actually define correct usage (see:
https://en.m.wikipedia.org/wiki/List_of_language_regulators). Thus, there are actually NO rights and wrongs in English, just conventions we follow in the quest to make our thought clear.

(Go look up Oxford comma jokes to see my point)
I, on the other hand, don't actually care; I just find it humorous to point out this info to people who are self-proclaimed grammar nazis yet have errors in their own posts elsewhere in a thread

(and I promise I meant all of the above in good fun

).