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.

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    $\begingroup$ I believe at present the set is empty. Next year its cardinality may change. $\endgroup$ – user856 Aug 25 '15 at 3:53
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    $\begingroup$ There are certainly modal logics that implicitly deal with shifting truths depending on contexts. Been too long since I took modal logic. $\endgroup$ – Thomas Andrews Aug 25 '15 at 3:59
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    $\begingroup$ If using standard set theory (ZFC), the only items that can be element of a set are other sets. Assuming the 2016 Olympics gold medallists will not be sets (which I expect to be the case), they cannot be members of a ZFC set. If not using ZFC, it depends on the set theory you use. $\endgroup$ – celtschk Aug 25 '15 at 7:01
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    $\begingroup$ I have voted to put this question on hold because it is a question of philosophy, not of mathematics. It is related to the problem of "future contingents" described by Aristotle - is the future set in stone, but unknown, or is the future unset until it occurs? That is not a mathematical question. $\endgroup$ – Carl Mummert Aug 25 '15 at 10:28
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    $\begingroup$ @Dan Christensen: the issue is not with whether we know what elements are in the set; the issue is that the law of the excluded middle for set membership, combined with a set definition referring to the future, implies that future events are already determined, which is a point that has been heavily debated in philosophy, and which is not a mathematical question. $\endgroup$ – Carl Mummert Aug 26 '15 at 10:51

(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.

  • $\begingroup$ That are perhaps good reasons hereabouts why those interested in sets of mathematical objects should just get on and construct their theory of them without wondering if and how it can be extended to cover non-mathematical sets. Sure. But that isn't a reason to suppose that there aren't non mathematical sets. $\endgroup$ – Peter Smith Aug 25 '15 at 8:19
  • $\begingroup$ It is a great reason. Just like there are no non-mathematical real numbers either. $\endgroup$ – Asaf Karagila Aug 25 '15 at 8:24
  • $\begingroup$ What if you replace 2016 with 2216? It is highly unlikely that Gold medal winners of the 2216 Olympics (assuming we do have one) are alive today (although, who knows!). Does that change the answer in any way? $\endgroup$ – Masked Man Aug 25 '15 at 8:57
  • $\begingroup$ @MaskedMan Yes, you certainly can't use my reasoning about that case. I suppose one view could be that it is currently the empty set, gets populated as the relevant people get born, and then eventually becomes the empty set again. But on some views, the empty set is a mathematical fiction, and on that view, there is no such set yet as the Gold medal winners of the 2216 Olympics: though there will be one, we hope. $\endgroup$ – Peter Smith Aug 25 '15 at 9:04
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    $\begingroup$ I think that the original question is one of philosophy. Without trying to start a long discussion here, the question seem to me to ask whether a predicate about the state of events in the future can be said to have a well defined extension in the present. It seems related to future contingents. The point made here that a set cannot "change" its elements is important, though. $\endgroup$ – Carl Mummert Aug 25 '15 at 10:30

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

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    $\begingroup$ There is a key difference between questions that we simply don't know the answer to, versus questions about the future. The difference is that most people accept that the fact that $\pi$ is irrational is a permanent aspect of $\pi$: $\pi$ has always been irrational. This is a common feature of mathematical questions. On the other hand, it is a long debated point in philosophy whether future events are already determined, but unknown, or whether they are genuinely undetermined until they occur. This is a very well-debated question in philosophy. $\endgroup$ – Carl Mummert Aug 26 '15 at 10:14

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 $X$.

  • $\begingroup$ So you have conclusively argued that the question should be closed as "not clear what you are asking"? $\endgroup$ – Thomas Aug 26 '15 at 11:33
  • $\begingroup$ I think the question is clear, but the answer comes down to philosophy, rather than to mathematics. Indeed, as soon as we start using axiomatic set theory, this entire issue disappears. @Thomas $\endgroup$ – Carl Mummert Aug 26 '15 at 12:22
  • $\begingroup$ Which axioms of ZFC, if any, are not satisfied by the supposed set of Olympic gold-medalists in next year's games? $\endgroup$ – Dan Christensen Aug 26 '15 at 13:18
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    $\begingroup$ @Dan Christensen: axioms are satisfied by a model, rather than by a single set. But, in any case, "the set of Olympic gold medalists in next year's games" isn't written in the formal language of ZFC. If you could translate it into that formal language for me, or at least sketch how you propose to formalize it, we can check whether the comprehension scheme lets us construct it. (Recall that all objects in ZFC are sets, not people.) The issue in the original question is entirely for informal set theory carried out in natural language, not with axiomatic set theory, where such issues don't arise. $\endgroup$ – Carl Mummert Aug 26 '15 at 13:50
  • $\begingroup$ Not sure what you mean by a model in this context, but from set theory, we can infer that for any set S, there will always be more subsets of S than there are elements of S. It doesn't matter if S is the empty set, the set of coins in my pocket, the set of fish in the sea or the set of Olympic gold-medalists in next year's games. We don't have to formalize the notion of pockets, fish or Olympic gold medals to apply the above notion in these cases. Or so it would seem to me. $\endgroup$ – Dan Christensen Aug 26 '15 at 14:28

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.

  • $\begingroup$ Well, yes, but I think the a/the issue for the OP is that the fact that this set is empty now, but maybe not so later, which is different from the properties that a classical set has. And I am not the down-voter, BTW. $\endgroup$ – Gary. Aug 25 '15 at 17:44
  • $\begingroup$ Don't see why people downvoted. To such a vague question, there is surely more than one reasonable answer. $\endgroup$ – D_S Aug 26 '15 at 2:16
  • $\begingroup$ In ordinary set theory, the members of a set cannot change over time - a set is simply what it is, regardless of any definition we might try to put on it. If the elements of a set could change, then it would no longer be the same set, because a set is determined by its collection of elements. $\endgroup$ – Carl Mummert Aug 26 '15 at 10:17
  • $\begingroup$ @CarlMummert I understand that you want to drag this question into logic, but as a "working mathematician" I would just define a time dependent set as a map from $\mathbb{R}$ to sets. I'll update my answer. $\endgroup$ – Thomas Aug 26 '15 at 11:38
  • $\begingroup$ I think this is the easiest and most natural way to deal with the issue, from the point of view of ordinary mathematics. $\endgroup$ – Carl Mummert Aug 26 '15 at 13:54

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