Statistical Inference — MLE consistency for a pathological p(theta)

Let $X$ be a random variable which assumes only two values 1 and 0, representing occurrence of success or failure in an experiment with only two outcomes. The probability of getting success in the experiment is $p(\theta)$ defined as: $p(\theta)=\theta$ if $\theta$ is rational, $1-\theta$ if $\theta$ is algebraic irrational. Define an estimator $T=\frac{\sum_{i=1}^n x_i}{n}=$ Sample proportion of success. Which one of the following is correct?