There is more to this question than just the fact that we do not know who will win medals at the 2016 olympics. The question is actually philosophical. But, rather than addressing the philosophy, which is arguably off-topic here, I will discuss two more mathematical issues, although they are still related to philosophy.
1. Definitions in naive set theory
First, it is commonly thought that mathematical facts are permanent. For example, even though we do not know whether $\pi$ is a normal number, it is generally believed that its status will not change. There is something different about future Olympics. Suppose for the sake of argument that "the set of Olympic medalists in 2016" defines a set. Suppose that someone, Joan, is in that set today in 2015. Does this mean that Joan has been predestined, since the beginning of time, to win a medal? That would be a controversial philosophical position, saying that future "contingent" events are already predestined to occur.
One common response to this predestination question is that the meaning of "the set of Olympic medalists in 2016" might change over time. (Some people try to say that the members of the set will change, but this response is what they mean.) This leads to the question of what sentences define sets in naive (natural language) set theory. In general, a phrase that defines a set must have the property that, even if we do not know, it unambiguously assigns each possible element as a member or non-member of the set being defined. But one challenge in naive set theory is that there is no firm rule for which sentences define sets: the goal of having firm rules is what leads to axiomatic set theory.
If a particular phrase picks out one set $X$ now, but later picks out another set $Y$, then that phrase does not define a set. For example, "the set of integers that are the current hour on the clock" does not define a set, because the current hour changes each hour. By the same argument, "the set of medalists in the 2016 Olympics" seems to not define a set, unless the collection of medalists was already fixed forever - predestined. This is why the overall question is one of philosophy.
2. Translating philosophical issues into naive set theory
The overall question here is related to the problem of future contingents that dates back to Aristotle. Quoting from the Stanford Encyclopedia of Philosophy article just linked:
Central to the discussion in this famous Aristotelian text is the question of how to interpret the following two statements:
- “Tomorrow there will be a sea-battle”
- “Tomorrow there will not be a sea-battle”
Aristotle considered questions like: Should we say that one of these statements is true today and the other false? How can we make a clear distinction between what is going to happen tomorrow and what must happen tomorrow? (See On Interpretation, 18 b 23 ff.).
The same issue appears with 2016 Olympic medalists: the philosophical question is whether we can say today in 2015 that a particular person certainly will, or certainly will not, win a medal in 2016.
There is a general way to take a philosophical question or paradox in natural language and turn it into an issue in set theory. In this case, we could make Aristotle's sea battle problem into a set theory problem by using the following "definition":
$X$ is a set that contains $0$, and contains $1$ if and only if there will be a sea battle tomorrow.
Assume for the sake of argument that the quoted sentence defines a set $X$. Now, from the ordinary mathematical viewpoint, the number $1$ either is or is not in $X$. The issue is not whether we know whether it is in $X$. The issue is that, if $1 \in X$ today, then it is already true today that there will be a sea battle tomorrow, and if $1 \not \in X$ then it is already true today that there will not be a sea battle tomorrow. Either way, if we agree that $1$ either is or is not in the set, it seems that we agree that what will happen tomorrow is already determined today, if we accept definitions like the one that was used to make $X$.