In so many different instances we need to be able to construct sets of functions. The first axiom of set theory (at least in the order that I learned) says that $a\in b$ is only a proposition if a and b are both sets. Thus if a relation (and furthermore a function) is an element of a set then it too must be a set by this axiom. This is fine because half of the time we describe relations as subsets of some Cartesian product $X\times Y$. The other half, though, we say relations are predicates of two variables, i.e. $R(x,y)$ is true for certain $x$ and $y$ and false for others. If we combine these descriptions, we get what I call the "set-predicate correspondence axiom" that says if $a$ is an element of $b$ then $b(a)$ is true and if a is not an element of $b$ then $b(a)$ is false. In other words, sets are predicates though not necessarily the other way around. This is the only way I could think of to allow constructions of sets of predicates such as relations, functions, etc.

Has this axiom already been established? Do we at least accept this idea in mathematics? If not, how do we compensate for the discrepancy between two definitions of relations and the desire to construct sets of functions (or other predicates) as elements?

  • $\begingroup$ No; predicates are linguistic entities. They are part of the mathematical language of our theory: theory of numbers, theory of sets, etc. Sets are mathematical objects: they are what the theory of sets speak about. $\endgroup$ – Mauro ALLEGRANZA May 31 at 8:30
  • $\begingroup$ See also Comprehension principle. $\endgroup$ – Mauro ALLEGRANZA May 31 at 8:31
  • $\begingroup$ The common term for a collection defined by a predicate is a class, and a class which is not a set is a proper class. So your question is "Are sets also classes?" and the answer is yes. This was discussed before as well. $\endgroup$ – Asaf Karagila May 31 at 9:03
  • $\begingroup$ @AsafKaragila Actually my question was are sets predicates PRECISELY. We don't assume too much about the predicate $\in$, but if we assume sets are predicates $\in$ would just be the predicate where $(x\in P)\Leftrighrarrow (P(x))$. $\endgroup$ – Cam White May 31 at 16:00
  • $\begingroup$ The term "predicate" is syntactic. Sets are semantic. Formally, no. Sets are not predicates for the same reason humans are not helium atoms. But given a set, we can define a predicate, using a parameter, which will define exactly the set we started with. $\endgroup$ – Asaf Karagila May 31 at 16:02

Yes, each set $x$ corresponds to a unary predicate $\varphi_x(y) := y\in x$ which is a projection of the membership relation, which is a binary predicate $\varphi_\in(x,y) := x\in y.$

And yes, while sets correspond to predicates, not all predicates correspond to a set. For instance, $\{(x,y): x\in y\}$ is not a set. Collections of sets that do not correspond to a set are called proper classes.

(Note, I've used "set-builder" notation here, which is really "class-builder" notation, to describe classes. In set theories like ZFC where we don't have a formal notion of class, classes just correspond to predicates. For instance, $\{x:\varphi(x)\}$ just corresponds to the predicate $\varphi(x).$ And since we have an ordered pairing function, one doesn't really need to distinguish between predicates of different arities: $\{(x,y): x\in y\}$ can either be thought of as a binary predicate $x\in y$ or as a unary predicate "$z$ is an ordered pair whose first element is $x$ and second element is $y$ and $x\in y.$)

Similarly, many other logical functions and relations are proper classes, like the collection of all sets , the function $F(x,y) = \{x,y\},$ etc. But sometimes, like in the case of $X\times Y =\{(x,y): x\in X\land y\in Y\}$ we can show from the axioms that there is a set whose elements are all sets satisfying the predicate.

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  • $\begingroup$ Yes I mentioned that not all predicates correspond to a set. $\endgroup$ – Cam White May 31 at 2:11
  • $\begingroup$ We all know there's a correspondence. Would you say that sets ARE predicates or is that not something that can be said? $\endgroup$ – Cam White May 31 at 2:13
  • $\begingroup$ @CamWhite Informally, yes. Formally, no. Sets are objects. A set's extension (the class of all objects left-related to it by $\in$) is a predicate. But intuitively we just think of sets as being their extensions... as the axiom of extensionality tells us that sets and extensions are one-to-one, and of course intuitively sets are thought of as collections. $\endgroup$ – spaceisdarkgreen May 31 at 2:16
  • $\begingroup$ So formally, a predicate cannot be an element of a set? $\endgroup$ – Cam White May 31 at 2:47
  • $\begingroup$ @CamWhite Right. The elements of sets are just the sets in its extension, i.e. they are sets (assuming we're working in a system like ZFC where everything is a set). Predicates are a completely different kind of thing. For instance, in ZFC they are treated as logical formulae, i.e. part of the system we use to talk about sets, not part of the universe of sets. You might want to research the keyword 'metatheory'. $\endgroup$ – spaceisdarkgreen May 31 at 3:14

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