Context is importantGiven a system that contains a contradiction; that system can prove anything (is inconsistent).

The system of axiomatic mathematics contains contradictions. Therefore the system of axiomatic mathematics is inconsistent.

Axiomatic mathematics argues that inconsistency doesn't apply to the whole system. Specifically, a subset Z of an inconsistent system Y can be considered as a wholly distinct system, and thereby Z can be considered (potentially) consistent. (please read response to Gwydion, below)

{You might like to consider THE inconsistent system. An inconsistent system can prove anything. Proving a statement within a system means that statement is part of the system. So... All inconsistent systems contain every statement. As such, all inconsistent systems are equivalent. They all contain every possible statement. We can thus refer to The inconsistent system; which just happens to be the system of all possible statements.}

Description/emulationTake an assumed consistent formal System A containing a number of statements that have been proven true with respect to the axioms. We take three of these statements and (preserving their truth value) use them as the axioms for System B. So - axioms x, y and z are true statements within the context of System A. And asserted true axioms of system B.

System B is a subset of System A. Anything we can prove in System B can also be proven in System A.

If we happen to prove that System B is inconsistent; then System A must also be inconsistent.

Suppose, instead, that we alter the truth value of one of the statements. Either we assert that one of the axioms is false (just as reasonable as asserting the axioms true), or we take the inverse of a statement (e.g. !x) and assert that axiom to be true. (We'll assume !x is true).

!x is obviously false with respect to System A. Were !x to appear as true in System A alongside x is true; then System A would contain a contradiction and therefore be inconsistent.

In this case, System B's axioms of !x, y and z cannot possibly come from System A.

Changing the truth state of axioms based on an existing formal logical system means you are discarding the formal logical system in favour of... what?

So - describing or emulating an axiomatic system means taking statements with a known context and using them as the basis for defining/describing/emulating another system.

Gwydion wrote:When you remove an axiom from a set (creating a subset), some statements that were previously true/false may remain the same but others may become undetermined. Therefore, it is not valid to assume any true statement in the superset will remain true in any/all subsets. This applies directly to the concept of the PoE - once you remove the inconsistency from your axioms, you can no longer "prove" all statements true in the same way.

I'll restate your last argument then: We have a set of statements which is a system in its own right, in addition to being a subset of a larger system that is inconsistent. Within the superset, we can prove every sentence, so any conclusions drawn about this subset within the superset are meaningless. If this subset is consistent, then we can still make logically meaningful inferences within it; if the subset is also inconsistent, then we are back where we started.

{Thank you for your input. You are clearly trying to understand what I have misunderstood and bring this mistake to my attention (you are being constructive in your criticism).}

Context is important.

Given an inconsistent system, our context is that inconsistent system. Within that inconsistent system we can prove that every statement is contradicted by itself. Remove any statement from System A and every other statement still allows us to invoke the PoE.

We know enough about the inconsistent system to know that within the context of that system, we can not only prove anything, but we can claim any statement means anything we want. A set of axioms taken from an inconsistent system could each have any or all conceivable meanings. This, after all, is the reason for axioms. We need a known starting point.

I'm not trying to say that a proof within an inconsistent axiomatic system must apply to every other axiomatic system. I'm pointing out that once a system is inconsistent everything is lost. Including any conceivable path back to some known set of axioms.

@Madaco: System 1 is inconsistent if whatever describes System 1 is inconsistent. The phrase "The sky is orange" is fine by itself. The problem arises when we try to insist that the phrase is definitely either true or false but not both and not some inbetween (or off to the side) state.

I would suggest that System 4 is a null system. Non-existent. As such, I am hesitant to try and define properties of 'nothing'.

Regarding Map/territory: If the map is sufficiently detailed then it may not be possible to distinguish between the map and the territory. This is what mathematics strives for - descriptions that can be treated as if they were the object itself.

Hopefully a close reading of the main section above will clarify what I mean by description/emulation with respect to an axiomatic system - and hopefully that coincides with the standard mathematical use of description.

Regarding (mathematical) philosophy: Nothing so subtle. My aim is to show that any system that contains the concept of inconsistency must itself be inconsistent.

@LaserGuy: Two contradictory proofs within an axiomatic system prove inconsistency - not consistency. Indeed, given that the argument is that all axiomatic systems are inconsistent; I would expect you to be able to prove anything - including that a given axiomatic system is consistent. Proof of inconsistency trumps all other possible proofs.

@Flumble: I sympathise with your viewpoint. However, throwing out a result because you don't like it isn't very rigorous mathematics. The fact that axiomatic mathematics contains singularities that give absurd results should, by itself, be a warning that there are deeper problems within axiomatic mathematics (No matter that it appears possible to isolate the singularities).

Moreover, knowing that we can prove anything with the inconsistent system, and that the inconsistent system contains all possible statements allows us to prove that starting from the set of all possible statements there is no possible way to construct a consistent system (Brief summary: start with the inconsistent system. Create a path to a supposedly consistent system by (e.g.) removing an axiom/statement at a time. Apply the known rules of the inconsistent system (not the supposed rules of the destination system - that system isn't the context we start with)).

SummaryA system cannot describe itself. In order to describe a system we need some other known system to describe a set of known axioms.

Hence we use natural language to describe our first axiomatic system and then use the formally defined systems to describe other formal systems.

For all of axiomatic mathematics; System A provides the context for understanding the set of axioms for System B.

Where System A is an instance of the inconsistent system; we cannot describe anything (or we cannot know what it is that we are describing) (The inconsistent system can prove any meaning of any statement).

Where System B is inconsistent, then System A is also inconsistent (See LaserGuy's last paragraph). If System A ever describes any instance of the inconsistent system, then every system described by System A is inconsistent (from the context of System A).

Change the truth value of a statement within System A and you have discarded System A. Whatever you assert beyond that point is arbitrary and has no relationship with mathematics.

Can we change the truth value of a statement in natural language and still preserve the known state of axioms described by natural language?

It doesn't matter.

If natural language is capable of describing both consistent and inconsistent systems, then natural language is inconsistent - and every system from that point on is also inconsistent.

"But natural language is exempt from all the rules...".

The Principle of Explosion applies if it

can apply. Can we apply the PoE to natural language? Yes.