It's interesting to note Wigner's drawing attention to the unreasonable ineffectiveness of mathematics outside of physics
I think you make a mistake, confusing axioms, and 'atomic rules'. To understand the modern use, we should look at how it evolved, from Euclidean geometry, where it's axioms were seen as 'self evident' elementary propositions. Geometry was considered the fundamental strata of mathethematics at least until Newton's time:
"Newton was convinced that only geometrical (as opposed to algebraic) proofs can be considered certain, and indeed he recast even the mathematics of Principia in geometrical garb (using the synthetic method of fluxions). Favoring geometrical techniques was part and parcel of his ideal of injecting certainty into natural philosophy; in this he saw himself in opposition to the “skeptical probabilistic” attitude of many members of the Royal Society (such as Robert Hooke and Robert Boyle)." - from a review of Guicciardini's book on Newton
Essentially Euclidean axioms are assumptions, and modern mathematics like the development of alternative geometries from the 19th century on, revealed that there are alternative sets of these.
"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 and are often shown in symbolic form (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 most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., parallel postulate in Euclidean geometry). 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), and there may be multiple ways to axiomatize a given mathematical domain. - from Wikipedia on axioms
It's notable that many historical proofs relied implicitly on commutation, but non-commutative mathematics has proven essential in quantum mechanics. I take the first type of axiom to be definitional, the second to be assumptions.
The axioms of general relativity have been given as
General relativity can be constructed from the following principles:
- The Principle of Equivalence
- Vanishing torsion assumption (∇XY−∇YX=[X,Y])
- The Poisson equation (or any other equivalent Newtonian mechanics
-as discussed here
This has a specific aim, reducing the assumptions to a minimum and providing a basis to demonstrate consistency of deductions of the system with the axioms, without producing contradictions.
The axioms of Quantum-Field theory are still disputed.
There are issues for the axiomatising method in general posed by Godel's results, which show there are consistent sets of principles which cannot be recursively axiomatised, ie. which statements are theorems cannot be determined by an automatisable method. Godel Incompleteness puts a fundamental limit on what can achieved with axiomatising, ending this part of the objectives of the Hilbert programme and of the Principa Mathematica. Stephen Hawking clearly stated he believed Godel's results made a Theory Of Everything impossible.
The most convincing account of how mathematics gets results about the world is I think Nancy Cartwright's How The Laws Of Physics Lie. We make abstractions, and deduce results which can only be as reliable as the abstractions are valid.
I would return to geometry to understand what abstractions are. We can see how symmetries provide economy in describing things: a sphere can be described with two numbers and greatly simplifies calculating the moments of inertia and centre of mass of a body, say.
The Bekenstein Bound shows us there is a maximum amount of entropy possible in a space, and that this occurs with blackholes. This means they are the most disordered possible system, the least economy can be achieved using symmetries. This contrasts with the 'no hair' theorem, and has led to the holographic principle and suggested extension of the principle of conservation of information to a universal law (& presumably like all conservation laws, there is an associated dimensional symmetry). Look at Conformal Cyclic Cosmology, it is suggested when the universe has decayed into only photons, they don't experience time, and by geometric arguments this is equivalent to a Big Bang, or whitehole. It can be described by photon energy-density only at that point.
So, we have systems of simplified explanation, in which we seek to have minimum assumptions, and no self-contradictions. Will there be alternate systems? Clearly, like the different systems of geometry. I would suggest what is happening is a fractal process of increasing disorder between these 'absolute' information states, with emergent complexity found in systems with fractional dimensions - like the embedding of our 4D space in a 5D one in the holographic principle. I would suggest these economies of explanation/account are not fundamental, but about emerging symmetries that represent relative order or complexity within the system, such as can preserved by biological systems consuming local Gibbs Free Energy, preserving a locally ordered system which would otherwise decay into a disordered one.
Economy of axioms as I see it in this picture, is like the attempt to reduce the fundamental constants, which is to say to understand tbe point on the universe timeline when it could be described with the greatest economy. One suggestion is that we can explain many fundamental constants as a fracture plane within the E8 hyperobject, which would reduce our universe's initial conditions to a sponteneous symmetry-breaking event.
When you say
"otherwise the reality would be inconsistent"
what you really mean is, a situation would occur like the anomolous orbit of Mercury, or the Ultraviolet Catastrophe in modeling atoms - we would know our model lacked key qualities to account for inconsistencies. And we would amend the model, and reconsider the set of minimum assumptions, that we call axioms.