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.
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"?
"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.
Perhaps the more interesting question is whether some axiom can be proved using the other axioms, without that axiom in the system. A proof system can prove any of the axioms, since they are axioms.
And it is not a trivial question, a very long time went into trying to prove the parallel postulate from the other axioms of Euclidean geometry. Later a sound proof was provided that this could not be done.
Today we have better theory to know which axioms are redundant and which are not. Usually upon building an axiom system the redundant axioms are removed. However, this is not always the case, for example even in set theory the axiom of empty set can be either proved or left as an axiom.
Furthermore, we can't really prove that number theoretic systems are consistent without assuming even more axioms that can't be proven consistent. So ultimately it is possible that some of the axioms are still redundant and even cause inconsistency, even if almost no one intuitively believes this.
