Is an axiom a proof? From this comments discussion on Philosophy.SE:

"Check out formal logic resources - I'm not going to dig them out for you. Alternatively ask on Math.SE. An 'axiom is a proof' is a definition in formal logic - and not an axiom. In philosophical logic, you can dispute this - but then there you can dispute what counts as proof."

This comment did not align with the definitions I have always used in my head.  I have always treated an axiom as a statement that is assumed true without a proof, and a proof is a structure with which one proves the truth of a conclusion given the assumption that its premises are true.
As one descends deeper into formal logic, is there a school of thought where "an axiom is a proof" is actually a definition?  I haven't been able to find anything to support this in my searches on the internet, but the original poster of the comment is quite confident that searching will reveal said defintion.
 A: I guess both can hold, though this discussion has little to do with formal logics.
"An axiom is a proof" could be seen as a definition in a (meta)theory where you previously defined a proof. But without more restrictions, this merely creates an alias and is usually not done.
But in a theory where you defined both an axiom and a proof, with different meanings, you could adopt an axiom saying "an axiom is a proof".

In common mathematics, an axiom is a proposition accepted without a proof, whereas a proof is the logical deduction of a proposition from the available axioms. (The axioms are independent, you can't derive one from the others.)
A: In the usual formulation of a proof system, a formula $\phi$ is provable if either


*

*$\phi$ is an axiom of the proof system, or

*$\phi$ can be concluded from formulae $\phi_1,...,\phi_n$ that are provable in the proof system, using one of the proof rules of the system


Therefore we cannot say that "an axiom is a proof", since we have to say that a proof is a proof of something. But we can say that if $\phi$ is an axiom in our proof system, then the axiom $\phi$ constitutes a proof of $\phi$.
A: One can say that a proof of a statement $X$ is a finite sequence of statements where a) each statement in the sequence is either an axiom or follows from some previous staements via one of a handful of rules of inference and b) the last statement in the sequence is $X$.
- So if the statement $X$ is already an axiom, the one-term sequence with only term $X$ constitutes a proof of statement $X$. Hence if you do not distinguish between $X$ and the one-term sequence with only term $X$, tha claim is correct and an axiom fulfills the definition of a proof.
