Do all mathematical and logical axiomatic systems implicitly ground natural numbers?

Question

Maybe this question is more suitable for Philsophy SE, but I want to hear mathematicians' opinions.

Suppose that we have an axiomatic system $\mathcal{A}$ with axioms $A_1, A_2, A_3,\dots,A_n,\dots$ Notice that this at least implicitly grounds natural numbers, as $n \in \mathbb{N}$ is the only reasonable option. (Or is it? I'd love a counterexample to that, if anyone was crazy enough to try — if it is even possible — to construct an axiomatic system which somehow has a number of axioms which is not $0$, a natural number or infinity!)

Notice that even if we have some axiomatic system $\mathcal{B}$ which contains no axioms, its existence grounds the number $1$ (as the system itself, uncontroversially, is one), and therefore (all?) (a) natural number(s). (I added these parentheses because I'm not sure if natural numbers can be deduced solely from the fact that $1$ is implicitly grounded.)

This, as I see it, edges on mathematical Platonism, as some things (in this case, natural numbers), truly exist in the structure of any possible mathematical or logical system, even though they haven't been defined yet! Anyway, my questions boils down to this:

1. Is my observation philsophically and mathematically sound or is there a counterexample to my claim that the number of axioms can only be infinity, $0$ or a natural number?

2. Has any mathematician acknowledged this observation in his professional work?

Answer 1

I think you are confusing the axiom system with language used to describe the system. In your examples the latter language implicitly requires (some) natural numbers. That says nothing about the former.

In practice, I doubt that mathematacians would find much use for a system that was too weak to allow counting.

Answer 2

I would find the number of axioms in a system of rather minor importance compared to what the axioms actually imply. For example, any axiomatic system with a finite number of axioms is equivalent to a system with any other number of axioms (except 0), as we can add dummy axioms, or use conjunctions to reduce the number of axioms.

From the perspective of the thing that is being modelled by the axioms, there is not much inherently to say about the number of axioms that is being used to model it, except for it being impossible to find a finite axiomatisation or not.

Nothing about e.g. "Group theory" says that it should have a theory with 4 axioms. The situation inside the thing being described by the axioms has not a lot to do with the way it is described by those axioms.

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As to your claim that an axiom system can only have a natural number or infinitely many axioms, this of course depends on what you understand as "number". Note that this is a discussion in the meta discourse, so not a discussion inside the theory that we're describing with our axioms.

Usually the number of things is interpreted using cardinality. As long as we're working in a logic where the formulas are denumerable (such as FOL), it is indeed impossible to find a "number" of axioms that is not one of those options.

This is because of how "number" is defined, namely two collections have the same number of things if they are bijectively relatable to each other. When your language is denumerable, this automatically implies that the number of axioms you can build is a cardinal number.

Note that even if you somehow work in a logic that is not denumerable, then still the only exception to sets of axioms not having a cardinal number as size, would be infinite collections.