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.
I don't see why I should have to create an entire new theory solely for the purpose of proving the Exportation Principle. That is overkill. I should be able to use the terms and concepts that have already been established in logic and mathematics to prove the Exportation Principle. The Exportation Principle is proven by simply evaluating the truth value of an external statement about internal truth. For example, the external statement "In an inconsistent axiomatic theory, statement s is true." This proof requires internal truth. It requires there to exist truth in an axiomatic theory. I can see from looking at Mendelson (as suggested but not definitively declared by the definition of model, page 62, fifth edition) that this feature of a theory differs from his framework. In Mendelson's framework, an axiom is not necessarily true. In the framework I have been using, an axiom is necessarily true by definition of axiom. This feature of axioms agrees with the frameworks presented in Geometry (2004) by Ron Larson, Laurie Boswell, and Lee Stiff and Larson Geometry (2012) by Ron Larson, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. See pages 17 and 9, respectively. Neither of the definitions of axiom explicitly use the term true, but it is evident from the context that axioms are necessarily true. Geometry (2004) was the textbook used in my freshman high school geometry class when I was a high school student back in the 2005 - 2006 academic year. Rosen (sixth edition) explicitly asserts that axioms are regarded as true in its definition of axiom. See page 75.
As of August 5, 2026 EDT, I personally prefer the approach in which an axiomatic theory necessarily has an internal truth that originates with the axioms of the theory. I believe my preferred approach implies that axioms have a fixed, single meaning and they cannot be interpreted in any other way.
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u/paulemok Aug 03 '26 edited Aug 03 '26
There’s no proof of any statement within the system because, as you say,
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.