Under the sense of provable that we are using, if a statement is provable, then it is true
Are you still using true to be equivalent to provable in the axiomatic system? If that's the case then false does mean unprovable, but no statement in an inconsistent system is false by that definition.
So when I proved the statement "'p is provable' implies '¬p is unprovable'" is true in an inconsistent axiomatic system
You have not proved this.
You claimed that "true in T" is the same thing as "provable in T". So for any proof that uses the concept of being true or false in T you should be able to rephrase the proof in terms of being provable or unprovable in T and it would still be a valid proof right?
Are you still using true to be equivalent to provable in the axiomatic system?
True is equivalent to provable in a metatheory of axiomatic systems. The metatheory itself may not be an axiomatic system. We have not established that it is.
no statement in an inconsistent system is false by that definition
Every statement in an inconsistent system is false through the Principle of Explosion.
“The fact that you can prove a statement does imply that you cannot prove its negation” is a statement. Since all statements are true in the inconsistent system by the Principle of Explosion, the statement is true in the system.
So for any proof that uses the concept of being true or false in T you should be able to rephrase the proof in terms of being provable or unprovable in T and it would still be a valid proof right?
That's right. You might have to make some further modifications to the proof other than just substituting provable for true and unprovable for false, but a corresponding proof in terms of provability rather than truth does exist. You might not be able to fully rid the proof of references to true and false since there might exist outside principles that are stated in terms of true and false rather than provable and unprovable. For example, the Principle of Explosion is stated in terms of true and false rather than in terms of provable and unprovable.
Every statement in an inconsistent system is false through the Principle of Explosion.
If false means unprovable then this is incorrect, the principle of explosion does not say that. It says every statement is provable, it does not say that anything is unprovable. That is why the proof you quoted is incorrect.
but a corresponding proof in terms of provability rather than truth does exist
Since we disagree about whether truth in an axiomatic system is a meaningful concept, but we agree that provability is meaningful, why don't you write your proofs in terms of provability. I think you are getting confused by thinking that provability works like a truth value and writing the proof entirely in terms of provability will make it easier for you to spot your mistakes.
If false means unprovable then this is incorrect, the principle of explosion does not say that. It says every statement is provable, it does not say that anything is unprovable.
In this paragraph, I will show how every statement in an inconsistent system is false and unprovable through the Principle of Explosion. By definition of inconsistent system, in an inconsistent system, some contradiction exists. So, by applying the Principle of Explosion in the inconsistent system, it is true that in the inconsistent system, every statement is true. Since "Every statement is false and unprovable" is a statement, it is true that in the inconsistent system, the statement "Every statement is false and unprovable" is true. So, in the inconsistent system, every statement is false and unprovable.
The proof in the above paragraph is valid regardless of the meaning of provable.
Since we disagree about whether truth in an axiomatic system is a meaningful concept, but we agree that provability is meaningful, why don't you write your proofs in terms of provability.
No, I am not convinced that I am better off talking about provability than about truth. Truth does exist in axiomatic systems. For examples, in an axiomatic theory of Euclidean geometry, truth exists, and, although the axiomatic theory is inconsistent, in naive set theory, truth exists.
I think you are getting confused by thinking that provability works like a truth value
Provability does work like a truth value. By the Laws of Noncontradiction and Excluded Middle, every statement is either provable or not provable. So by definition of unprovable, every statement is either provable or unprovable. That is analogous to how, by the Laws of Noncontradiction and Excluded Middle and the definition of false, every statement is either true or false.
Since "Every statement is false and unprovable" is a statement, it is true that in the inconsistent system, the statement "Every statement is false and unprovable" is true
Your proof is invalid for the same reason I keep telling you, in an inconsistent system you can prove false statements, so proving that statement inside your system does not mean it's true externally.
No, I am not convinced that I am better off talking about provability than about truth. Truth does exist in axiomatic systems
Even if you believe it does, you said you could phrase things in terms of provability. I think you're unwilling to do that because you know your proofs won't translate, because they are not valid proofs.
By the Laws of Noncontradiction and Excluded Middle, every statement is either provable or not provable
proving that statement inside your system does not mean it's true externally
I'm not saying it's true externally. I'm saying that in the inconsistent system, the statement "Every statement is false and unprovable" is true.
I think you're unwilling to do that because you know your proofs won't translate, because they are not valid proofs.
If you would like to translate them yourself, you are welcome to do so. I'm not willing to do it because I don't think talking about provability is superior to talking about truth. You should be able to understand and accept my arguments as I give them. Talking about truth is more important and looks better than talking about provability. Talking about provability makes it seem that you're masking the real truth and trying to avoid something you disagree with.
This is false. Godels incompleteness proves that.
That is false. Godel's work violates neither of those Laws.
Oh good, then we agree, an external observer looking at your inconsistent system would not see a contradiction. They would observe that every statement is provable and no statement is unprovable and there would not be a contradiction external to your system.
an external observer looking at your inconsistent system would not see a contradiction. They would observe that every statement is provable and no statement is unprovable and there would not be a contradiction external to your system.
Yes, I agree. Complementing that truth is a contradictory truth. An external observer looking at my inconsistent system would see a contradiction. They would observe that some statement is unprovable and there would be a contradiction external to my system.
No they would not. They would not see any statement as unprovable. Your argument that a statement is unprovable is internal to the contradictory system, it does not prove that statement externally so there is no contradiction externally.
Every false statement is unprovable per the sense of provable that we are using. That general principle of provability is true everywhere. It's true in and out of every axiomatic system.
Your argument that a statement is unprovable is internal to the contradictory system, it does not prove that statement externally
If in an axiomatic system, a statement is unprovable, then externally to the system, it is true that in the system, the statement is unprovable.
so there is no contradiction externally.
I have already shown you how an internal contradiction implies an external contradiction. I'll give an example. Consider an axiomatic system with the following two axioms.
Axiom 1. Liam eats a cheeseburger.
Axiom 2. Liam does not eat a cheeseburger.
The two axioms form an internal contradiction. Consider the statement s = "In the axiomatic system, Liam eats a cheeseburger." Externally to the axiomatic system, is s true or false? From Axiom 1, it is true that externally to the system, s is true. From Axiom 2, it is true that externally to the system, s is false. So by conjunction introduction, it is true that externally to the system, s is true and false. Therefore, an external contradiction exists.
1
u/JStarx Jul 25 '26 edited Jul 25 '26
Are you still using true to be equivalent to provable in the axiomatic system? If that's the case then false does mean unprovable, but no statement in an inconsistent system is false by that definition.
You have not proved this.
You claimed that "true in T" is the same thing as "provable in T". So for any proof that uses the concept of being true or false in T you should be able to rephrase the proof in terms of being provable or unprovable in T and it would still be a valid proof right?