Question: This question was inspired by [this](https://math.stackexchange.com/questions/111428/lemma-vs-theorem) question. 

Does it really make sense to say that a formal system has some number of axioms, say three, or ten, etc? E.g., take a formal system that admits conjuction elimination and has axioms $ A\_1, A\_2,..., A\_n $. But a formal system that has the single axiom $ A\_1\wedge A\_2\wedge ...\wedge A\_n $ is essentially the same formal system!

So, is the concept of number of axioms in a formal system subjective, just as there is no objective difference between a lemma and a theorem?

**P.S.** I'm indeed talking about finitely axiomatizable formal systems only. I.e. when I say "axioms", I mean it - no axiom schemata allowed.

Answer: 
Certainly if I give you a formal system with an explicit list of axioms, say 42 of them, then it makes sense to say that the system has 42 axioms. It's hard to dispute that.

But you make a good observation, that there many *equivalent* formal systems with all sorts of different numbers of axioms. To be precise, say two formal systems (using the same symbols) are equivalent if they prove exactly the same theorems. It's often useful consider formal systems *only up to equivalence*, when what we care about is the theorems they prove, not their presentations.

So the question is really about whether by formal system you mean "explicitly defined formal system" or "formal system up to equivalence". The former has a well-defined number of axioms, the latter does not.
If you step out of boolean logic and look at multivalued logics, there are places where having a conjuction (for example in BL (<http://en.wikipedia.org/wiki/BL_%28logic%29>) if you conjunct the axioms with what is known as strong conjunction) is different from having the axioms separately (i.e. the sets of deductions won't match up properly).

In the boolean case, the number of axioms (I'm talking about finite axiomatizations as asked in the question) doesn't really matter if you are interested in the theorems. But the derivation of a proof (in a formal system) will be different. However there is good reason to split up the axioms and talk about them separately; the human mind just deals with that approach better.