I, too, will take the mathematical route. According to https://en.wikipedia.org/wiki/Axiom:
As used in mathematics, the term axiom is used in two related but
distinguishable senses: "logical axioms" and "non-logical axioms".
Logical axioms are usually statements that are taken to be true within
the system of logic they define (e.g., (A and B) implies A), while
non-logical axioms (e.g., a + b = b + a) are actually substantive
assertions about the elements of the domain of a specific mathematical
theory (such as arithmetic). When used in the latter sense, "axiom",
"postulate", and "assumption" may be used interchangeably. In general,
a non-logical axiom is not a self-evident truth, but rather a formal
logical expression used in deduction to build a mathematical theory.
As modern mathematics admits multiple, equally "true" systems of
logic, precisely the same thing must be said for logical axioms - they
both define and are specific to the particular system of logic that is
being invoked. To axiomatize a system of knowledge is to show that its
claims can be derived from a small, well-understood set of sentences
(the axioms). There are typically multiple ways to axiomatize a given
In both senses, an axiom is any mathematical statement that serves as
a starting point from which other statements are logically derived.
Within the system they define, axioms (unless redundant) cannot be
derived by principles of deduction, nor are they demonstrable by
mathematical proofs, simply because they are starting points; there is
nothing else from which they logically follow otherwise they would be
classified as theorems. However, an axiom in one system may be a
theorem in another, and vice versa.
Let's take logical axioms first. What would be the value in defining "god" to be the set of (true, self-evident) axioms within the system those axioms define? Yes, doing so would enable to one to answer the question "What is god?" but the answer seems sterile and unlikely to appeal to many.
With regard to non-logical axioms, such as the axioms of arithmetic, the axioms are merely starting points for the deduction of additional statements or sentences. Of course, one could define "god" as this set of axioms, but, again, what would be the value? The same questions some people raise about "god", they could now raise about the axioms, e.g., Are the axioms "true"? Are the axioms "consistent"? Are the axioms "complete"? It's not obvious what one would gain by this.
A few comments on Goedel. Among other things, he's showed that:
There are an endless number of true arithmetical statements which
cannot be formally deduced from any given set of axioms by a closed
set of rules of inference. (Goedel's Proof, Nagel and Newman, New
York University Press, 1958, p.98)
This suggests that rather than equating "god" with the axioms that define a given mathematical system -- and which are part of that system -- it may make more sense to equate "god" with the set of all true statements that cannot be deduced from those axioms -- which are not part of that system.
All this presumes though, the validity of the mathematical analogy to begin with. On that and other matters, I agree with @CortAmmon.