As of August 2015, is the "set" of all gold medalists in the 2016 Olympics a set?

Question

As of August 2015, is the "set" of all gold medalists in the 2016 Olympics a set? I think it is since the defining property is very clear. However, given any $x$, we do not know if $x$ is in this "set" at the moment.

I want to compare my example with one provided in the answer: the set of rational numbers is clearly a set. Assume our math knowledge is limited so that we don't known whether $e$ is rational or not. It seems that being unable to determine membership itself can't be a reason for not being a set.

Answer 1

Think of this example: long time ago we did not know if $\pi$ or $e$ is in the set of rational numbers.

Answer 2

(1) Note that all 2016 Olympic winners are currently existing people (you can't win an Olympic medal aged one!). We don't know who they are. They might be Usain, Jessica, ... or they might be Justin, Katerina ... Assume for the moment, that sets can have non-mathematical objects as members. Then the set {Usain, Jessica, ...} exists -- it is a set of currently existing people, and exists now. The set {Justin, Katerina ...} exists -- it is a set of currently existing people, and exists now. There are lots and lots of other such sets. One of them, unknown to us now, happens to be the set of people who will be 2016 Olympic winners [assuming we've fixed on some determinate meaning for that]. Or, if you take an Aristotelian line on the "open future", you'll prefer to say: it isn't yet fixed which set that is, and it will only later become true of one of the sets of people that currently exists that it is correctly describable as the set of winners. Either way, we will have to wait and see which one that turns out fit the description. Still whoever the winners will be, the people in question exist right now, and there is -- if we hold that in addition to any things that exist right now there is a set containing just them -- a set of them.

So it is wrong to say of this case "I believe at present the set is empty. Next year its cardinality may change". If you believe in concrete sets, you might indeed allow sets whose membership changes as their members go in and out of existence. On this view, the membership of the set of people who are (or in 2016 become) Olympic winners will change, some time after the Games, as its members begin to die off. But that point isn't germane here. Irrelevant sci-fi fantasies apart, those people live and breathe right now, and taken together, whoever they are, they form a set right now. If only I knew a more helfpul description of that set, I'd be off to the bookmakers today.

(2) Should we however say, as some do, that sets can't have non-mathematical objects as members? As opposed to (say) Kunen who allows {C}, the singleton of a cow (he just doesn't want to talk about such things, rather than deny they exist). Or (say) Halmos who allows sets of wolves, grapes or pigeons, but who says they are not the concern of set theory. Or (say) Potter who thinks such impure sets are the concern of set theory and whose preferred theory is a version with urelements (i.e. there can be, at the bottom level of the hierarchy, objects which are members of sets which aren't themselves sets -- where these objects can be physical objects).

There are reasons why mathematicians may ignore sets which have non-mathematical objects as members (though physicists might be interested). Indeed, there are reasons why set theorists go on to ignore any sets which have non-sets as members (once they see how to model other mathematical objects in a universe of pure sets). But those aren't good reasons to deny the existence of sets which have non-sets as members. We refer all the time to collections, sets, classes -- when in Cantor's famous words, there is a "gathering together into a whole of definite, distinct objects of our perception or of our thought". If you want to hijack the word "set" for purely mathematical purposes, so be it. Call what we used to call sets of concrete objects "classes" or "collections". There remain all the questions there were before about the logic and metaphysics of such talk: we'll still want a theory of such things. And to answer questions like whether there is now a class of 2016 Olympic winners.

Answer 3

One way to circumvent the logical implications outlined in the other answers would be to define a time dependent set as a map $\mathbb{R} \to \text{Sets}$. Then the object you define could be modeled as a time dependent set $$GM2016(t) = \{\text{people who have won a gold medal at the 2016 Olympics until time $t$}\}.$$ which is a subset of the time dependent set $\{\text{all people who have been alive until time $t$}\}$.

So $GM2016(\text{August} 2015) = \emptyset$. I expect particularly drastic changes next summer with only occasional updates (doping, etc.) afterwards.

Answer 4

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 $X$.