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Even though ZFC is a first-order theory, it can interpret properties that are not first-order in a different language. For instance, in the language of graphs, the property of a graph being finite is not first-order expressible. But in any model of ZFC, we can find a set corresponding to the set of all finite graphs.

Yet how can we know that every mathematical concept can be interpreted in ZFC? Could there not exist some property or object in the literature which can not be represented in a model of ZFC? Do we just work from the assumption that ZFC can interpret everything -- or are there good reasons to believe so?

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    $\begingroup$ Isn't it the case that the category of groups Grp can't be represented in ZFC, because you would want the objects (groups) to form a set, and in ZFC there is no set of all groups. To make category theory work in ZFC you have to restrict Grp to only talk about the groups in some particular universe of discourse. But you can do category theory without this restriction and category theorists often do. $\endgroup$
    – MJD
    Commented Jan 30 at 15:03
  • $\begingroup$ What does $ZFC$ stand for? $\endgroup$
    – Dominique
    Commented Jan 30 at 15:10
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    $\begingroup$ @Dominique - set theory: Zermelo-Fraenkel axioms plus the Axiom of Choice, or $\mathsf {ZFC}$. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Jan 30 at 15:12
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    $\begingroup$ Or: en.wikipedia.org/wiki/ZFC . Also possibly useful: Common acronyms used on main and meta $\endgroup$
    – MJD
    Commented Jan 30 at 15:21
  • $\begingroup$ It can't, interpret everything. Proper Classes, sets of Dogs, donkeys, etc... Whether it can interpret everything within the domain of mathematics is something for current and future mathematicians to decide. It really depends on what we consider within our realm of mathematical discourse. Arguably, the existence of independence results for things like CH is some evidence that ZFC actually isn't "interpreting" CH correctly. If you believe that the Continuum Hypothesis Should be provable. $\endgroup$
    – Michael Carey
    Commented Jan 30 at 15:39

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Your question may be conflating two distinct notions of first-order, which may contribute to your puzzlement. First, a system such as Peano Arithmetic is a first-order theory in the sense that its axioms only quantify over "numbers". Similarly, you can have statements over $\mathbb R$ that quantify only over elements thereof (rather than subsets of $\mathbb R$, for example). That would be the first sense of "first-order". A claim that most of mathematics could be interpreted this way would indeed be very novel (and implausible).

On the other hand, there is a much vaster sense of "first-order" when we speak of ZFC, in the sense that it can quantify over any elements of the cumulative hierarchy, such as for example over elements of the traditional superstructure. That's a lot of interpretive power, indeed. Certainly, finiteness (and finite graphs) can be handled in ZFC; this simply means a correspondence with an element of $\mathbb N$ (which is defined as the least inductive set).

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