How do we know the true value of a parameter, in order to check estimator properties? For example, we say that an estimator is unbiased if the expected value of the estimator is the true value of the parameter we're trying to estimate. However, if we already know the true value of the parameter, why are we trying to estimate it? I know there's something I'm missing here, like some theoretical true value that we can find, otherwise I don't see how we can check for unbiasedness, consistency etc.

Consider i.i.d. sample $X_1, \ldots, X_n \sim N(\mu, 1)$. We assume that all samples are from normal distribution with mean $\mu$, which is unknown, and known variance $1$.
Now, let $\bar X = \frac{1}{n} \sum_{i=1}^n X_i$.
Then $E(\bar X) = \mu$.
So no matter what $\mu$ is, $E(\bar X) = \mu$. It is not necessary to know what $\mu$ is.
