# The Matrix in ZFC: A Set-Theoretic Foundation of Matrices?

Today, we can use ZFC to found all of mathematics. We show from ten or so axioms the existence of everything from ordered pairs and Cartesian products to relations and functions (whose existence follows from the existence of Cartesian products). I have learned, from reading books like Enderton's 1977 Elements of Set Theory, that it is not enough simply to define something (for example, we define the ordered pair of $a$ and $b$ as the set $(a,b)=\{\{a\},\{a,b\}\}$), but we must also prove that what we have defined exists within ZFC.

I have done some searching, but have not found a set-theoretic construction of matrices. Wikipedia's Matrix page states the definition of an $m \times n$ matrix as simply a "rectangular array", but this definition does not show that a matrix is a set which actually exists in ZFC.

Does anyone know how to construct matrices in ZFC? Please feel free to point me in the direction of books in set theory which tackle this. Thanks for reading my question!

• Infinitely many axioms, actually: two of them are axiom schemata, not single axioms. – Brian M. Scott Feb 24 '15 at 1:09
• If you already know how to construct ordered pairs, cartesian products, functions, and so forth in terms of ZFC, then why are you having trouble with matrices? Why do you think you need to search for a reference? Why can't you just make up your own encoding? – DanielV Feb 24 '15 at 1:15
• There's no fault in asking how something is traditionally defined. – Mathemanic Feb 24 '15 at 1:48
• What about categories ? – Rene Schipperus Feb 24 '15 at 3:56

Consider an $m\times n$ matrix with elements in some field $F$. Let $J_m=\{1,2,\ldots, m\}$ and $J_n=\{1,2,\ldots, n\}$, then a matrix can be viewed as a function $a:J_m\times J_n\to F$ with $a(i,j)=a_{ij}$.
• Sorry. I don't have a reference. At some point I realized that subscript notation is an alternate form of function notation. This idea works for sequences. For example, a sequence of real numbers can be considered as a function $\Bbb N\to\Bbb R$. From there it's not too hard to generalize this to matrices. – Tim Raczkowski Mar 18 '15 at 11:43