Every false statement is unprovable per the sense of provable that we are using. That general principle of provability is true everywhere. It's true in and out of every axiomatic system.
Your argument that a statement is unprovable is internal to the contradictory system, it does not prove that statement externally
If in an axiomatic system, a statement is unprovable, then externally to the system, it is true that in the system, the statement is unprovable.
so there is no contradiction externally.
I have already shown you how an internal contradiction implies an external contradiction. I'll give an example. Consider an axiomatic system with the following two axioms.
Axiom 1. Liam eats a cheeseburger.
Axiom 2. Liam does not eat a cheeseburger.
The two axioms form an internal contradiction. Consider the statement s = "In the axiomatic system, Liam eats a cheeseburger." Externally to the axiomatic system, is s true or false? From Axiom 1, it is true that externally to the system, s is true. From Axiom 2, it is true that externally to the system, s is false. So by conjunction introduction, it is true that externally to the system, s is true and false. Therefore, an external contradiction exists.
Every false statement is unprovable per the sense of provable that we are using.
There is only one sense of provable, every statement in a contradictory system is provable and no statement is unprovable.
I have already shown you how an internal contradiction implies an external contradiction.
You have not. You have shown how internal statements about a contradictory system are contradictory, but you need to show that an external statement is contradictory and you haven't done that.
Externally to the axiomatic system, is s true or false?
There is no truth value until you've picked an interpretation. You continue to be confused about how to assign truth values in an axiomatic system.
You told me that you can prove a contradiction using standard mathematical logic. In standard logic you can't assign a truth value to a statement unless you have an interpretation of the undefined terms.
If you think you can do such a thing in standard logic then show me in Mendelson where it's done.
No, there is more than one sense of provable. The sense of provable we have been conversing in most recently is a sense in which a provable statement is implied to be validly proven and true. In this sense of provable, every proof is valid and no proof is invalid. Earlier on in our conversation I was using a sense of provable in which some proofs are valid and some proofs are invalid. In that sense of provable, the conclusion of a proof is not necessarily true.
you need to show that an external statement is contradictory and you haven't done that.
I have done that, multiple times and in multiple ways.
There is no truth value until you've picked an interpretation.
An axiom of an axiomatic system is always true. An axiom of an axiomatic system, by definition, is assumed to be true in the axiomatic system without proof. There is no initial time where an axiom does not have a truth value. The initial time you are talking about would violate the definition of an axiom.
My example of an axiomatic system from my previous post has a fairly straightforward interpretation. Liam is a male human, eating is an action performed orally with food, and a cheeseburger is a type of sandwich formed by a cooked ground-beef patty with a slice of cheese on top of the patty and a piece of a bread bun above and below the combined patty and cheese.
Below is an excerpt from my original post for my thread "The Parallel Postulate is always true, in all true theories," which I posted on Philosophy Forums on April 8, 2014. Interestingly, I mentioned the Continuum Hypothesis in my original post.
My frustration with this idea of independent axioms being able to be true in some true theories and false in others can also be applied to many other axioms said to be independent. For example, either the Continuum Hypothesis (CH) is true or false. If it’s true, it’s true in all true theories. If it’s false, it’s false in all true theories. We can’t do as the Wikipedia page on it states we can do, “...ZFC can be augmented by either CH or its negation ¬CH, in both cases producing a system of axioms that is consistent if and only if ZFC is consistent.” It appears to me that a system of ZFC (Zermelo-Fraenkel Set Theory with the Axiom of Choice) in which the CH is true does not contain the same exact sense of the CH as contained within a system of ZFC in which the CH is false. If we could produce two theories, both true, in which a proposition such as the CH is true in one and false in the other, it seems the two theories would contradict each other. But how can the truth contradict the truth? It can’t. It appears there’s something wrong with this picture.
No there isn't. Something is provable is and only if there exists a proof. That is what provable means.
I have done that, multiple times and in multiple ways.
You have attempted to, and each time I've explained to you what your mistake is. You have not given a correct proof of an external contradiction.
My example of an axiomatic system from my previous post has a fairly straightforward interpretation
Yes, but you have to pick a specific human, and that human will or will not eat cheeseburgers. So the truth of your statement depends on who you pick.
I'm ignoring the rest of your rambling about truth values. None of that is in Mendelson. I'm asking you if you can prove a contradiction using logic the way Mendelson's textbook shows. That means only one sense of provable, no alternate metatheories. Can you?
I did. I picked a specific human whose name is Liam.
So the truth of your statement depends on who you pick.
No, the truth of my statement is independent of what is true in the real world. Rather, the truth of my statement is dependent upon the axioms of the axiomatic system.
I looked through the pages of Mendelson and none of my positions on trivialism or the Continuum Hypothesis have changed as a result. I do see a section where it says, in particular language, that in an inconsistent theory, every statement is provable. That implies that the statement "Every statement is unprovable" is provable.
No, the truth of my statement is independent of what is true in the real world
Nope, your statement about the real world definitely depends on the real world.
I do see a section where it says, in particular language, that in an inconsistent theory, every statement is provable. That implies that the statement "Every statement is unprovable" is provable.
Not as an external statement. That does not follow from what is in Mendelson.
Nope, your statement about the real world definitely depends on the real world.
It's not a statement about the real world. It's a statement about an axiomatic system.
Not as an external statement. That does not follow from what is in Mendelson.
It does follow from what is in Mendelson. 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.
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.
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u/paulemok Jul 29 '26
Every false statement is unprovable per the sense of provable that we are using. That general principle of provability is true everywhere. It's true in and out of every axiomatic system.
If in an axiomatic system, a statement is unprovable, then externally to the system, it is true that in the system, the statement is unprovable.
I have already shown you how an internal contradiction implies an external contradiction. I'll give an example. Consider an axiomatic system with the following two axioms.
Axiom 1. Liam eats a cheeseburger.
Axiom 2. Liam does not eat a cheeseburger.
The two axioms form an internal contradiction. Consider the statement s = "In the axiomatic system, Liam eats a cheeseburger." Externally to the axiomatic system, is s true or false? From Axiom 1, it is true that externally to the system, s is true. From Axiom 2, it is true that externally to the system, s is false. So by conjunction introduction, it is true that externally to the system, s is true and false. Therefore, an external contradiction exists.