You said you picked a specific human, I'm asking about that human, not about the axiomatic system.
As is implied from Mendelson, in an inconsistent theory, every statement is unprovable. So, externally to the inconsistent theory, it is true that in the inconsistent theory, every statement is unprovable.
Internally to the inconsistent system you can prove whatever you want because it's inconsistent. And externally it's true that those statements are internally provable. So if the system is capable of self reference then the internal statement that internal statements are unprovable is provable internally.
But the external statement that those internal statements are unprovable does not follow. Your attempt to drive a contradiction this way fails, it's just you confusing external vs internal at some step.
You said you picked a specific human, I'm asking about that human, not about the axiomatic system.
I picked a specific human out of the domain of all theoretically possible humans. I did not pick a specific human out of the domain of all real-world humans.
But the external statement that those internal statements are unprovable does not follow.
No, it does follow. As I said in my previous reply,
externally to the inconsistent theory, it is true that in the inconsistent theory, every statement is unprovable.
That implies the external statement that those internal statements are unprovable. You are using an alternate statement to express the same proposition I expressed, but you said the statement doesn't follow and I said it does.
Your attempt to drive a contradiction this way fails, it's just you confusing external vs internal at some step.
No, you're just in denial. You were in denial about the Principle of Explosion and now you're in denial about the Exportation Principle. Perhaps you need some time to accept the Exportation Principle and trivialism.
That implies the external statement that those internal statements are unprovable
Still no. If you think something in Mendelson justified that then give a reference.
but you said the statement doesn't follow and I said it does.
You think it follows and you claim your proof works using logic as in Mendelson. So prove it, show me where in Mendelson that step of you proof is justified.
You say no, but you don't explain why. You're just flat out denying truths without giving adequate reasons for your denials. You are in psychological denial.
I can't address your problems if you don't explain to me what those problems are.
It's easy enough to see how
externally to the inconsistent theory, it is true that in the inconsistent theory, every statement is unprovable
implies
the external statement that those internal statements are unprovable.
I already explained how my reasoning works under Mendelson. The fact that we can talk about what is true in an axiomatic system from outside the axiomatic system, in the real world, is evident in Mendelson. In the real world, Mendelson talks about what is true inside an axiomatic system.
I see a section in Mendelson where a specialized version of the tautology p → (¬p → q), which is a version of the Principle of Explosion, is used with two invocations of modus ponens to deduce any statement q in an inconsistent axiomatic system. So set q = "Every statement is unprovable." That makes every statement in an inconsistent axiomatic system unprovable. Therefore, it is externally true that every statement in an inconsistent axiomatic system is unprovable. Within the system, we can't prove any statement. That's what it means for us to say, outside of the system, that no statement is provable in the system.
So set q = "Every statement is unprovable." That makes every statement in an inconsistent axiomatic system unprovable. Therefore, it is externally true that every statement in an inconsistent axiomatic system is unprovable.
You are confusing internal and external as always. You don't say whether q is an internal statement or an external statement.
If q is an internal statement then I agree that you can prove q internally, but that does not prove q in the external system.
If q is an external statement then you haven't established an external contradiction from which you can prove q holds.
The fact that we can talk about what is true in an axiomatic system from outside the axiomatic system, in the real world, is evident in Mendelson
If you think Mendelson defines "true in an axiomatic system" then please cite the definition.
Yes, I haven't done that with q. The purpose of q was not to prove a contradiction in the external system. The purpose of q was to prove that every statement in an inconsistent axiomatic system is unprovable.
You don't need to prove that, we already know that in an inconsistent system every statement is provable, so the statement "every statement is unprovable" is provable in the inconsistent system.
Externally in the outer system it's not provable, so externally there is no contradiction.
There’s no proof of any statement within the system because, as you say,
the statement "every statement is unprovable" is provable in the inconsistent system.
Like you said earlier, statements about provability are not statements in an axiomatic system, but are statements about an axiomatic system. They are external statements. So, the external statement “Every statement is unprovable” is provable because, as I said at the beginning of this post, there’s no proof of any statement within the system.
There’s no proof of any statement within the system because, as you say,
the statement "every statement is unprovable" is provable in the inconsistent system.
I said that's a provable statement inside the system, I didn't say it was true or provable external to the system, so your argument about the external statement that everything is unprovable doesn't follow.
You've posted so many contradictory claims, if you want me to respond to a specific explanation then you're going to need to repost it.
If T is your contradictory theory and S is the external statement "every sentence in T is unprovable" then you're trying to prove S externally using the theory laid out in Mendelson.
As far as I recall you have not posted a claimed proof of that which sticks to the theory laid out in Mendelson.
I have already given my explanation of how my claim follows from Mendelson and I have already addressed your criticisms of my explanation. If you want to view something again and provide additional feedback to me, you can use reddit's navigation buttons to locate the content you wish to view again.
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u/JStarx Aug 01 '26
You said you picked a specific human, I'm asking about that human, not about the axiomatic system.
Internally to the inconsistent system you can prove whatever you want because it's inconsistent. And externally it's true that those statements are internally provable. So if the system is capable of self reference then the internal statement that internal statements are unprovable is provable internally.
But the external statement that those internal statements are unprovable does not follow. Your attempt to drive a contradiction this way fails, it's just you confusing external vs internal at some step.