Decidability of Turing Machine accepting exactly 14 words Would you say that the following problem is undecidable? 
$$L_1 = \{\langle T \rangle \mid T \text { accepts 14 words}\}$$
My intuition says that this must be undecidable, and I want to try to reduce it from the emptiness problem for Turing Machines. If we can tell whether or not a TM accepts 14 words, then we should be able to tell if it accepts 0. Since we know EMPT is undecidable, I have been trying to do reduce EMPT to this problem, but I am not sure exactly how to do it. 

Can anyone confirm if this is actually a valid starting point, and if it is, any hints on how to formulate a formal reduction?
Your intuition is correct. Probably you can reduce emptiness (or maybe its complement) to this language but it's not the way I'd do it. Instead, think about how you prove that emptiness is undecidable.

You want to know if $M$&nbsp;halts on input&nbsp;$w$ so you produce a TM that accepts everything if $M$ accepts&nbsp;$w$ and rejects everything if $M$ rejects&nbsp;$w$. A slight modification of that proof will show that your language is undecidable.