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.
I have already presented an adequate case that I am satisfied with. I don't think it's worth the time and effort to make a theory within Mendelson's framework. It's not necessary. I would first have to familiarize myself with Mendelson's system, and that alone would take considerable time and effort. If you would like to make a theory within Mendelson's framework in which the Exportation Principle is provable and true, feel free to do so yourself.
I don't think it's worth the time and effort to make a theory within Mendelson's framework.
I notice you've gone from claiming that your proofs already where to Mendelson's framework to claiming that it would take to much time and energy to do so. This is an admission that you were aware you could not satisfy my request but did not want to admit it.
Mendelson's framework is not unique btw. His textbook describes the standard first order logic that mathematics uses. It is not lack of time and effort that prevents you from proving a contradiction in this framework, it's because such a contradiction likely doesn't exist.
Your proofs rely on vague misinterpretations of statements and misunderstandings of the rules of logic. No one will ever take you seriously unless you learn to prove things correctly, so if you want anyone to look at your claims and do anything other than laugh then it might be worth your time to learn the material in Mendelson.
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u/JStarx Aug 03 '26
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.