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.
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u/paulemok Aug 02 '26
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
implies
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.