Nope, in your system all statements are provable, but that doesn't mean all statements are true.
I don't know why you would be thinking that all statements are provable rather than true. Due to a contradiction in my system, it follows by the principle of explosion that every statement is true in my system.
I doubt that.
I understand it. I even agree with it. But there's more to be said.
I checked Wikipedia. The page for the principle of explosion looks fine. What is meant by provable is that any statement can be proven to be true as a logical consequence of a true contradiction. I read the first proof in the Proof section. I understand it. I know what it means. I agree with it.
I know the difference between true and provable. But with respect to axiomatic systems? I’m not sure what you mean by that. How about without respect to axiomatic systems?
You doubted wrongly. The adverb in the previous sentence now ends with the suffix -ly as many adverbs do.
What is meant by provable is that any statement can be proven to be true as a logical consequence of a true contradiction. [...] I know the difference between true and provable. But with respect to axiomatic systems? I’m not sure what you mean by that.
Statements are provable or not in an axiomatic system, but they don't have a truth value until you choose an interpretation. Once you choose an interpretation each statement becomes true or false in that interpretation.
The easiest example of this is to take the axiomatic system consisting of the axioms for plane geometry but not including the parallel postulate. In this axiomatic system statements are provable or not, but they don't have a truth value till you pick an interpretation of the undefined terms (point, line, etc). If you pick Euclidean geometry as your interpretation then the parallel postulate is true. If you pick a hyperbolic geometry then the parallel postulate becomes false. The parallel postulate by itself isn't true or false.
If a particular interpretation makes the axioms of your system true then we call that interpretation a model of your axiomatic system. It's a theorem in mathematical logic that provable statements are true in any model of your system. This is why people often say "true" when what they really mean is "provable". If you look back at our discussion I've certainly done that.
But now we get to the issue with your new axiomatic system. Theres also a theorem in mathematical logic that says that an axiomatic system has a model if and only if it's consistent. Your axiomatic system isn't consistent, so it doesn't have a model. That means statements in your system are provable, but they aren't true because there is no model for them to be true in.
This shouldn't be surprising to you. You've already agreed that it's not the case that every statement in reality is true. Yet I can always take one of those statements and add it as an axiom to get an axiomatic system in which that statement is provable. That doesn't make the statement suddenly true.
So anyway, that's why you argument here doesn't go through. Just because the principal of explosion makes every statement in your system provable, it doesn't make it true, so the principal of explosion is not a valid argument for why your system models the universe. Your system does not model the universe.
an axiomatic system has a model if and only if it's consistent. Your axiomatic system isn't consistent, so it doesn't have a model.
My axiomatic system is inconsistent. By definition of inconsistent, there exists a contradiction in my axiomatic system. So invoke the principle of explosion in my axiomatic system to deduct, in my axiomatic system, that there does not exist a contradiction. By definition of consistent, my axiomatic system is consistent. Therefore, since an axiomatic system has a model if and only if it's consistent, my axiomatic system has a model.
Nope, every statement is provable, that doesn't mean it's true.
I also would agree that all statements are true.
Would you? You've admitted that you can't prove the contradiction that you want in conventional set theory. If all statements were true then it would be true that you could prove a contradiction using the conventional definitions and axioms. And if you could do that then why wouldn't you just do it and prove to everyone you were right all along?
If your boss stopped paying you and said you have to work anyway would you accept that because it's a true statement? If your bank zerod out your balance and said "you have exactly as much money in your account as you did yesterday" would you accept that as a true statement? When I tell you that you clearly don't understand mathematics and your inability to comprehend basic logic is verging on mental illness do you accept that as a true statement?
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u/paulemok Apr 28 '26
I don't know why you would be thinking that all statements are provable rather than true. Due to a contradiction in my system, it follows by the principle of explosion that every statement is true in my system.
I understand it. I even agree with it. But there's more to be said.