Suppose that there have been $n$ days and that the sun has risen on all of them. What’s the chance that the sun will rise tomorrow? Assuming that we start with a uniform prior on the probability that the sun will rise, then the chance [edit: this should say 'expected value of the probability'] is $$ \frac{n+1}{n + 2}$$ (See here for the details and relevant background assumptions.)
So if there have been $10$ days, the chance that the sun will rise tomorrow is $\frac{11}{12}$. Moreover, the chance that the sun will rise both tomorrow and the day after is $\frac{11}{12} \times \frac{12}{13} = \frac{11}{13}$.
Now let’s re-describe events: there have been $5$ double-days (a double-day is two normal days) and the sun has double-risen 5 times (the sun double-rises if it rises on two consecutive days). What’s the chance that the sun will double-rise again? Using the formula from before, it is $\frac67$. But $\frac67 \neq \frac{11}{13}$! What's going on here?
This is not really all that suprising, as a flat prior for the probability of the sun rising on a certain day is inconsistent with a flat prior for the probability of the sun double-rising on a certain double-day, given that the latter is given in terms of the former by $p_2=p_1^2$, so $f_{p_1}(p_1)=1$ implies $f_{p_2}({p_2})=\frac1{2\sqrt{p_2}}$.
What this shows is not so much a contradiction but the arbitrariness in choosing a flat prior for a variable in some representation when we could just as well choose a flat prior in a transformed representation. The general idea is that the differences are typically small (as in this case) for reasonable choices of variables and are quickly drowned out by the data, so they don’t matter that much in practical applications.
In the present case, the entire premise of a flat prior for something that seems to be governed by laws of nature is in any case questionable. Based on our overall experience of the law-like nature of much of reality, a realistic prior should emphasize probabilities near $0$ and $1$.