4 hours ago, studiot said:

6 hours ago, KJW said:

But if any axiom can actually be proven (or disproven) from the other axioms, then there are too many axioms, and the set of axioms is potentially inconsistent.

Surely this statement cannot be right.

Say there are G axioms and axiom G is found to be provable from the other axioms A to F.

Unless the proof of G is independent of one or more axioms, say B, how does this lead to an inconsistent set ?

The point is that if an axiom is proven or disproven, then it is no longer an axiom. If the "axiom" is proven, it is a theorem. If the "axiom" is disproven, the set of axioms is inconsistent.

Sometimes this is difficult to avoid. For example, in group theory, the existence of inverse elements can be expressed as:

For each element [math]g[/math], there exists an inverse element [math]g^{-1}[/math] such that [math]g^{-1} g = g g^{-1} = e[/math] (the identity element).

However, only one of the statements [math]g^{-1} g = e[/math] or [math]g g^{-1} = e[/math] is actually an axiom as the other statement can be proven. Which one is the axiom, and which one is the theorem is an arbitrary choice, so it seems natural to include both in a single statement so as to not arbitrarily break the symmetry.

Edited 19 hours ago19 hr by KJW

20 hours ago, Genady said:

cannot even be formulated in the weaker one

I have to admit that this actually took me by surprise. After taking some time to think about this, I realise that it is ironic that what I said about two different types of axioms, the "deeper stuff" that you said was "not even wrong", appears to be key to my misunderstanding of the notion of completeness. Yes, there are two different types of axioms: one that defines a mathematical universe, and another that constrains that universe. It would seem that I neglected the mathematical universe. That would be because of the way I view mathematics, which is that everything exists unless proven otherwise. That means, for example, I assume the existence of multiplication even if it has not been explicitly defined. My mathematical universe contains multiplication, contains infinity, contains transfinite numbers, contains the axiom of choice, the continuum hypothesis has a definite answer, etc. But of course, that's not how this subject in mathematics is done. The mathematical universe is defined explicitly by the axioms, and notions such as completeness are based on it. So, I actually can see how a system with few axioms can be complete. And I can see that Gödel's incompleteness theorem is not trivial.

Edited 9 hours ago9 hr by KJW

15 minutes ago, KJW said:

I have to admit that this actually took me by surprise. After taking some time to think about this, I realise that it is ironic that what I said about two different types of axioms, the "deeper stuff" that you said was "not even wrong", appears to be key to my misunderstanding of the notion of completeness. Yes, there are two different types of axioms: one that defines a mathematical universe, and another that constrains that universe. It would seem that I neglected the mathematical universe. That would be because of the way I view mathematics, which is that everything exists unless proven otherwise. That means, for example, I assume the existence of multiplication even if it has not been explicitly defined. My mathematical universe contains multiplication, contains infinity, contains transfinite numbers, the continuum hypothesis has a definite answer, the axiom of choice is true, etc. But of course, that's not how this subject in mathematics is done. The mathematical universe is defined explicitly by the axioms, and notions such as completeness are based on it. So, I actually can see how a system with few axioms can be complete. And I can see that Gödel's incompleteness theorem is not trivial.

I am really glad that this misunderstanding has been figured out. Thank you for clarifying its origins. I wondered and did not see where it comes from.