Are arithmetic operators really sets in $\mathsf {ZFC}$?

Question

Are two familiar arithmetic operators on $\mathbb N $, i.e. $+$ and $\cdot$, really functions in a set-theoretic sense? I.e., is '$1+2=3$' really '$\left ( \left ( 1, 2 \right ), 3 \right )\in+$'?

Answer 1

Yes. Everything is a set in $\sf ZFC$.

We define the natural numbers by recursion:

1. $0=\varnothing$,
2. $n+1=n\cup\{n\}$.

The successor operator is now defined on the natural numbers: $$S=\{(n,n\cup\{n\})\mid n\in\Bbb N\}$$

Using the successor operator we can define addition, and then multiplication, as usual by recursion. But we can also define these operations directly. By noting that $n+m=k$ if and only if $k$ is the unique natural number that there is a bijection between $k$ (as a set) and the set $n\times\{0\}\cup m\times\{1\}$; and that $n\cdot m=k$ if and only if there exists a bijection between $k$ and $m\times n$, if we write $|A|=|B|$ as a shorthand for the formula "There exists a bijection between the sets $A$ and $B$", then we can now write the operations:

$$\begin{align} +&=\Bigl\{((m,n),k)\mid |k|=|m\times\{0\}\cup n\times\{1\}|\Bigr\}\\ \cdot&=\Bigl\{((m,n),k)\mid |k|=|m\times n|\Bigr\} \end{align}$$

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It is worth pointing out that this is just one of many possible representations of the natural numbers and their operations in set theory. We can define many others, and it will not change their properties or behavior as we perceived the natural numbers before the interpretation (i.e. when not considered as sets).

The above have two clear advantages, though, the formulas defining the numbers and the operations are relatively simple, and these coincide and extend to ordinal and cardinal arithmetic when considering the von Neumann ordinals, and the $\aleph$ number assignment.

Answer 2

It depends on how you parenthesize the question:

[Are arithmetic operators really sets] in ZFC?

Yes, when we model mathematics in the formal theory ZFC, then the arithmetic operators will be represented by certain sets.

(However: Ordinal addition and multiplication are not sets).

Are arithmetic operators really [sets in ZFC]?

No, not really. People were adding and multiplying natural numbers for centuries before ZFC was invented, so it doesn't make good sense to assert that a particular representation in ZFC is what these operators "really" are. However suggestive the word "foundation" is, the role of ZFC is not to be all of mathematics, but to model the kind of mathematical arguments that real mathematicians actually accept.

What real mathematicians do is a great deal more intuitive and informal than what a cursory reading of an introduction to axiomatic set theory can make you believe. It just happens that "proofs that can be formalized in ZFC" turns out to be a pretty precise approximation of "proofs that most working mathematicians will accept as valid".

Of course the working mathematician's idea of which proofs are valid is to a certain extent shaped by knowledge of set theory, but only to a certain extent. Arguably the intuition behind the use of sets in mathematics outside set theory is more often close to some kind of type theory than to the one-sorted sets of ZFC.