Let $\phi$ be an undecidable statement of ZFC set theory, for example let's take continuum hypothesis.
What is the ontological status of the "set" $X=\bigl\{x\in\{1,2\}:x=1\text{ or }(x=2\text{ and }\phi)\bigr\}$ in ZFC?
Let $\phi$ be an undecidable statement of ZFC set theory, for example let's take continuum hypothesis. What is the ontological status of the "set" $X=\bigl\{x\in\{1,2\}:x=1\text{ or }(x=2\text{ and }\phi)\bigr\}$ in ZFC? 

closed as not a real question by Qfwfq, Noah Schweber, Steven Landsburg, Andrés E. Caicedo, Goldstern Jun 5 '13 at 2:30It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question. 


Although you've been given a hard time in the comments, I think that this is actually a serious question in the philosophy of mathematics, whose answer depends on one's philosophical position concerning the nature of mathematical truth. What is the nature of existence for the mathematical objects that we define? There are a variety of natural positions to stake out, so let me describe at least a few of them. There are of course many more; I mention several perspectives specifically on CH in my answer to Gil Kalai's question on solutions to the continuum hypothesis, which you may find relevant. As you describe the example, the puzzling thing here is that you seem to have defined a fairly simple set, yet we also find ourselves unable to answer some fairly simple questions about it, such as whether it has one or two elements. To my way of thinking, the philosophical question about the ontological status of your set is very similar to the philosophical question about the ontological status of the truth of $\phi$ itself; the two simply go together. So we could ask the question like this: When a statement $\phi$ is independent of ZFC, what is the ontological status of the truth of $\phi$? For example, does it still make sense to say that $\phi$ is definitely true or false, but definitely only one of these? Ontology has to do with the nature of realityin our case, mathematical realityindependently of, say, our knowledge about it. But provability and nonprovability (and thus independence) seem to have to do with our ability to know certain things, and thus relate to epistemological rather than ontological concerns. Let me describe a few of the philosophical positions that are commonly held about this kind of question. Traditional settheoretic Platonism. On the traditional Platonist view, sets exist in a unique real Platonic realmthe realm consisting of all setsand every settheoretic assertion has a definitive truth value in this realm. On this account, there is a fact of the matter about whether CH is true or false, or whether your assertion $\phi$ is true or false. On this account, the set you have described is either in fact identical to $\{1\}$ or is identical to $\{1,2\}$. The fact that $\phi$ is independent of the ZFC axioms merely illustrates our lack of knowledge about which one of these sets you have actually described. On this account, the set is definitely one of them or the other, depending on whether it is the case that $\phi$ is true or not, and exactly one of these is the case. The independence of $\phi$ is an irrelevant distraction from whether $\phi$ is true. On this view, the pervasive independence phenomenon is seen as a sideshow about our epistemological weaknessthe weakness of our theoriesrather than indicating any issue about the singular nature of mathematical truth. Formalism. On this view, there is no realm of real existence for mathematical objects like sets, and rather the mathematical process consists of the manipulation of sequences of symbols, such as the definition in your question. On this view, assertions of existence in mathematics are merely a way of speaking, and no actual existence of mathematical objects is being asserted. Formalism thus trivializes the ontology of the mathematical objects that mathematics is about. Plural realism, or the multiverse view. On the plural realist or multiverse conception of settheoretic truth, one holds that there are diverse concepts of set, each giving rise to its own settheoretic universe, a plurality of settheoretic worlds with their own settheoretic truth, but which can be connected in various ways, such as by forcing or by large cardinal embeddings. On this account, your sentence $\phi$ may be true in some of these worlds and false in other worlds, and your set will accordingly change its nature depending on the current settheoretic background concept of set under which it is being interpreted. You can read more about the multiverse perspective in set theory in some of my recent papers, especially "The settheoretic multiverse", RSL 2012. 


Consider the following descriptions of sets: (1) $X=\lbrace i\in\omega: i=0\rbrace=\lbrace0\rbrace$, (2) $Y=\lbrace i\in\omega: i=0\iff CH, i=1\iff \neg CH\rbrace=\lbrace 0\rbrace$ or $\lbrace 1\rbrace$, and (3) $Z=\lbrace i\in\omega: \text{ the $i$th Diophantine equation (in some natural listing) has an integral solution}\rbrace$. Now, the short version is: $ZFC$ (actually, much less than $ZFC$) proves that $X$, $Y$, and $Z$ exist. A slightly longer version, which might help you understand what's going on here: really, what I have is not three sets $X$, $Y$, and $Z$, but rather three firstorder formulas $\phi_1(x), \phi_2(x)$, and $\phi_3(x)$ with a single free variable. Now, for every such formula $\phi(x)$, $ZFC$ proves that $$ \lbrace i\in\omega: \phi(i)\rbrace$$ exists; this is just Separation + Infinity. There are really three potential issues here, and none of them have to do with the "ontological status" of (2).


