Axiomatic system and Proof for axioms

Question

So I am told by a friend that "axioms in an axiomatic system cannot be proved within the axiomatic system". I was wondering how true this is. Is there any actual mathematical theorem that states something like this.

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EDIT: Along the same lines, how true is it to say that "an axiom in an axiomatic system cannot be disproved within the axiomatic system"?

Answer 1

"axioms in an axiomatic system cannot be proved within the axiomatic system".

See Aristotle, Post.An, Bk.I, 82a7-82a9:

This is the same as to inquire whether demonstrations go on ad infinitum and whether there is demonstration of everything, or whether some terms are bounded by one another.

There are two uses of "proof" here: the usual one and the formal one.

In a formal system a proof is derivation in the system, i.e. a sequences of formulas where every formula either is an axiom or is derived from previous ones in the sequence by way of rules of inference.

The conclusion of the proof, i.e. the last formula in the sequence, is a theorem.

Thus, formally speaking, a one-line derivation, where necessary the formula is an axiom, is a formal proof whose conclusion, the axiom itself, is a theorem of the system.

But obviously in the common sense meaning of "proof", the above derivation will not be considered an "interesting" proof of the axioms:

A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference.

Proofs are examples of exhaustive deductive reasoning which establish logical certainty.