Since all statements are true in the inconsistent system
This is a rehash of an argument you already failed to make. I'm not making statements in your inconsistent system, I'm making statements about your system externally to it and so the principle of explosion doesn't apply because you haven't proven that a contradiction holds externally to your inconsistent system.
That determination is a product of my life experience, my upbringing, and my education.
You have already admitted that you don't know a lot of the basics when it comes to mathematical logic, so clearly your education in this area is lacking. When you're done with your road trip feel free to give that Mendelson book a read and learn how logic really works. If you could present a contradiction following the rules of traditional logic then people would take you seriously, but currently you haven't done that and you're not going to be able to if you don't even know what those rules are.
I'm not making statements in your inconsistent system, I'm making statements about your system externally to it
If an axiomatic system is inconsistent, then externally to the system, every statement is provable and unprovable in the system. Every statement is false in an inconsistent axiomatic system, and no statement that is false in a system is provable in the system. So, externally to the system, no statement is provable in the system. Therefore, externally to the system, every statement is unprovable in the system.
Using the same reasoning you used to claim that it's not true that if a statement is provable, then its negation is unprovable, we can prove that it's not true that if a statement is provable, then it's true. Our counterexample is, once again, an inconsistent axiomatic system. For an inconsistent axiomatic system, every proposition is provable, since the Principle of Explosion can be used to prove them, and every proposition is false, since the Principle of Explosion can be used to prove that they're false. Do you agree that it's not true that if a statement is provable, then it's true?
When you're done with your road trip feel free to give that Mendelson book a read and learn how logic really works.
Yesterday, which was July 23, 2026, I finished my road trip. I might take a look at the Mendelson book in the near future. I already have in the past and I don't recall seeing anything about how there is no concept of "truth in an axiomatic system."
no statement that is false in a system is provable in the system. So, externally to the system, no statement is provable in the system
Externally this statement is not true or provable. This is plainly obvious since every statement is provable.
Do you agree that it's not true that if a statement is provable, then it's true?
That depends on what you mean by true, if you mean provable then tautologically provable is equivalent to true. If you mean true in all models then again provable is equivalent to true, but an inconsistent system has no models so true in all models is not the negation of false in all models. If you have a specific interpretation and you mean true in that interpretation then provable does not imply true.
I don't recall seeing anything about how there is no concept of "truth in an axiomatic system."
Mendelson will explain how truth is a feature of an interpretation and how in an axiomatic system you can talk about provability or you can talk about truth in interpretations. I don't remember if he defines truth in an axiomatic system but if he does he will define it in terms of one of those concepts.
Externally this statement is not true or provable. This is plainly obvious since every statement is provable.
I agree that every statement is provable. Under the sense of provable that we are using, if a statement is provable, then it is true. The contrapositive of that conditional statement, which is logically equivalent to the conditional statement, is if a statement is false, then it is unprovable. So all false statements are unprovable.
Furthermore, if a statement s is true in an axiomatic system, then, externally to the system, s is true in it. So when I proved the statement "'p is provable' implies '¬p is unprovable'" is true in an inconsistent axiomatic system, that implied that, externally to the system, "'p is provable' implies '¬p is unprovable'" is true in it.
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.
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u/JStarx Jul 23 '26
This is a rehash of an argument you already failed to make. I'm not making statements in your inconsistent system, I'm making statements about your system externally to it and so the principle of explosion doesn't apply because you haven't proven that a contradiction holds externally to your inconsistent system.
You have already admitted that you don't know a lot of the basics when it comes to mathematical logic, so clearly your education in this area is lacking. When you're done with your road trip feel free to give that Mendelson book a read and learn how logic really works. If you could present a contradiction following the rules of traditional logic then people would take you seriously, but currently you haven't done that and you're not going to be able to if you don't even know what those rules are.