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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?

Alex Kruckman
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God bless
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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.

Ethan Bolker
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  • What is the difference between a language that describes a system and the system itself? Granted, syntax, the little black symbols on a piece of paper or a monitor aren't the abstract object they represent, but I think they resemble them accurately. My claim is that any axiomatic system, regardless how "weak" it is, grounds the existence of at least some natural numbers by its very nature. I don't see how a disparity between syntax and the real abstract mathematical objects weaken my proposition. – God bless Aug 05 '19 at 10:55
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    @GregorPerčič Of course there is a difference between the language describing a system and the system itself. I don't consider the theory of rings to be a ring itself. Actually mixing the system that is being used to express axioms in with the system being described by the axioms leads to severe problems with self-reference. Take for example Berry's paradox – Vsotvep Aug 05 '19 at 11:36
  • I agree that there is a clear ontological difference between a thing and its representation. However, I do not agree that the disparity is such that we cannot draw any conclusions about the thing from its representation. For example, if I write down a symbol, we automatically presuppose that this thing which the symbol represents, exists, or at least that we're considering its existence. This is a nice example of how we reason from representation to essence. My example is very similar: the symbol used to represent a thing is one, therefore we take it that the thing itself is one. – God bless Aug 05 '19 at 11:46
  • @GregorPerčič I think we will have to agree to disagree. When you start thinking about "essence" and "existence" you're no longer doing mathematics, or even mathematical logic. – Ethan Bolker Aug 05 '19 at 12:52
  • @EthanBolker Well, if metaphysical jargon bothers you, I can rephrase my claim. I think that it is nonsensical to claim that we can derive no information about what a certain thing is from its representation. The elementary maxim which I pointed out is that in order for a representation to exist, so must the actual thing it represents. My goal here is not to do mathematics or mathematical logic, I want to discuss the meta-reality underlying both (hence the tag 'philosophy'). – God bless Aug 05 '19 at 14:13
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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.

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.

Vsotvep
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  • Thank you for your answer! The point I wanted to hammer home is exactly that - and indeed, very misteriously - some things which are not yet defined are truly present in the things which are supposed to define them - natural numbers are an example of this, as I claim. Another example to illustrate my point would be the concept of existence: even though you can have a system in which the concept itself is not yet defined, it would be absurd to claim that the system doesn't exist. I'm intetested in these kinds of properties which cannot be defined, but can be conceived of only prima facie. – God bless Aug 05 '19 at 11:32
  • Well, trying to prove the existence of systems or natural numbers would need you to go a step up in the meta world of the language of your axioms. Of course, that implies that there is then a meta world that exists, but to show that, you will need yet another meta meta world. Why wouldn't it be turtles all the way down? – Vsotvep Aug 05 '19 at 11:35
  • Good point. It somewhat reminds me of Gödel's Incompleteness Theorems, which (roughly speaking) state that no system can be self-contained (please, correct me if I'm wrong). In other words, you will always have the "outer world", which you cannot reach or explore, yet it is necessary for those things which you can consider rationally. I've always had the greatest interest in the most fundamental things, so I apologise if this post is off-topic or is wasting anyone's time. I'm just in wonder... :) – God bless Aug 05 '19 at 11:40
  • It's definitely related. The first incompleteness theorem states that in a system that can do arithmetic, there will be statements that cannot be proved to be true or false. Interestingly enough, this is shown by introducing an unprovable statement using the meta discourse. The fact that a statement is unprovable, means that it must also be true (since there are no counterexamples), although this cannot be proved "from the inside". – Vsotvep Aug 05 '19 at 11:49