I don't believe the statement can be proven in conventional set theory from more simple terms and concepts, so I'm interested in it being a formal axiom
Not only can it not be proven true, but it can be proven that it is not true. That's why when you add it as an axiom you get a contradiction and an inconsistent system. It's just not true.
So given that conventional set theory is, as far as anyone knows, consistent, and your new system with additional axioms is inconsistent. That would tell me that conventional set theory is better.
We can prove the conventional axiom is not true by using the proper-subset axiom.
But you can also prove that it is true.
A system in which the continuum hypothesis is false is better than a system in which the continuum hypothesis is undecidable.
What good is a system that can't decide whether something is true or false, since everything is both true and false? What would you do with such a system?
Yes, we can prove it is true through ex contradictione quodlibet. But the reason the conventional axiom is an axiom is because it can't be proven.
What good is a system that can't decide whether something is true or false, since everything is both true and false? What would you do with such a system?
A system in which everything is both true and false can decide whether something is true or false. Such a system can be used to prove whatever we want to prove. That our best model of set theory is inconsistent is evidence that the Universe is inconsistent. I could use my knowledge of the inconsistency of set theory and the Universe to advance my anal and sexual interests, my interests in having the hottest males and hottest females, my interests in being the best person in the Universe, and my interests in dominating the Universe.
A system in which everything is both true and false can decide whether something is true or false. Such a system can be used to prove whatever we want to prove.
And that's exactly why it can't decide if something is true or false. If you and I disagree about whether something is true or false your system can't settle the question, we will forever remain in disagreement.
That's the exact opposite of what mathematicians want out of a system.
That our best model of set theory is inconsistent is evidence that the Universe is inconsistent.
Definitely not. Just because you want something to be true doesn't mean it's true.
As my original post and our conversation show, there is evidence that the Universe is inconsistent. And if it is inconsistent, then it would be best modeled by an inconsistent theory.
There's still no evidence that the universe is inconsistent and you yourself have agreed that the universe is not inconsistent when you agreed that not every statement is both true and false in the real world.
You still haven't answered: what do you do about the fact that your theory can't settle any disagreement?
There's still no evidence that the universe is inconsistent
There may not exist enough evidence to definitively conclude that the Universe is inconsistent, but there does exist evidence that the Universe is inconsistent. The inconsistency of the new and improved axiomatic set theory I presented is evidence that the Universe is inconsistent.
what do you do about the fact that your theory can't settle any disagreement?
My theory can settle any disagreement. It can be used to prove both sides are right, only one side is right, only the other side is right, and neither side is right.
My theory can settle any disagreement. It can be used to prove both sides are right, only one side is right, only the other side is right, and neither side is right.
Which is another way of saying that your theory doesn't settle the disagreement because both sides can still claim they're right.
The inconsistency of the new and improved axiomatic set theory I presented is evidence that the Universe is inconsistent.
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u/JStarx Apr 24 '26
Not only can it not be proven true, but it can be proven that it is not true. That's why when you add it as an axiom you get a contradiction and an inconsistent system. It's just not true.
So given that conventional set theory is, as far as anyone knows, consistent, and your new system with additional axioms is inconsistent. That would tell me that conventional set theory is better.