Logically inconsistent optimality condition I'm reading a book on monetary policy, and it defines the representative household's consumption as $$C_t = \int_0^1 C_t(i) di$$

so, basically a sum over the consumption of good $i$. 

Then later on, it says that the consumption of each good is identical $$C_t(i) = C_t(j) = C_t$$

This equation makes no logical sense. How can $C_t(i) = C_t$ for all $i$? $C_t$ is DEFINED as an integral sum over all those $C_t(i)$. They cannot be identical.

That's like saying 4 = 2 + 2, and then say 4 = 2.
There is $C_t$ the number and $C_t()$ the function. This is already so in your first equation. Your second equation says that the function is constant over $[0,1]$. In which case it is true that $C_t$ the number equals $C_t()$ the function's value times 1. 

The same thing with a different notation:
$$ C_t \triangleq \int_0^1 f_{C,t}(i) di.$$
If there exists a constant $c_t$, where for all $i$
$$f_{C,t}(i) = c_t.$$
then
$$ C_t \triangleq \int_0^1 f_{C,t}(i) di = \int_0^1 c_t di = c_t. $$