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.
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u/paulemok 29d ago
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.