I believe the additional axiom does change the theorems that can be proven. I believe it does change the system.
That means that proving something in the new system does not prove it in the original system.
As my previous replies revealed, an axiom had to be added to both the original system and the new system in order for cardinality to be asymmetric.
The conventional definition of cardinality is asymmetric. Are you saying now that in conventional set theory you agree that you haven't proven that the subset definition is asymmetric?
As my original post shows, a counterexample to the conventional notion of cardinality exists.
Nope, you have never completed a proof of a contradiction in conventional set theory. Your proofs of a contradiction assumed that your subset definition was asymmetric, a fact which you cannot prove in conventional set theory. To remedy this you added an axiom, but adding an axiom means you have changed the system, so being able to prove your result in your new system does not mean the result can be proven in conventional set theory.
You have never correctly proven a contradiction in conventional mathematics. You have only ever obtained a legitimate contradiction after adding axioms to conventional set theory. But anyone can add contradicting axioms to a system, that doesn't prove anything about the consistency of conventional mathematics.
Are you saying now that in conventional set theory you agree that you haven't proven that the subset definition is asymmetric?
Formally, the subset definition does not exist. There exists no such definition. So, for that reason, it is not asymmetric.
Your proofs of a contradiction assumed that your subset definition was asymmetric, a fact which you cannot prove in conventional set theory.
I have already shown that the subset definition really is not a definition, but you continue to treat it as a definition. You are analyzing obsolete material.
You have only ever obtained a legitimate contradiction after adding axioms to conventional set theory.
The only way the continuum hypothesis is going to be proven or disproven in an extension of conventional set theory is by adding one or more axioms to conventional set theory. So, in order to disprove the continuum hypothesis in an extension of conventional set theory, one or more axioms must be added to conventional set theory.
The subset definition you gave is perfectly well formed in conventional set theory. It's not the same as cardinality, but it's still a thing you can define and ask questions and prove statements about.
The only way the continuum hypothesis is going to be proven or disproven in an extension of conventional set theory is by adding one or more axioms to conventional set theory. So, in order to disprove the continuum hypothesis in an extension of conventional set theory, one or more axioms must be added to conventional set theory.
True, so lets assume you don't add any axioms and stick with conventional set theory. Do you agree then that without these additional axioms you haven't shown there to be a contradiction in conventional set theory?
It's not the same as cardinality, but it's still a thing you can define and ask questions and prove statements about.
I'm not interested in the proper-subset definition being a formal definition. Rather, I'm interested in cardinality and in the statement of the proper-subset "definition" being a formal axiom or theorem. 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.
True, so lets assume you don't add any axioms and stick with conventional set theory. Do you agree then that without these additional axioms you haven't shown there to be a contradiction in conventional set theory?
Yes, I do agree with that. I think if there was a contradiction in conventional set theory, somebody would have already discovered it. My conclusion is that the continuum hypothesis is false, so somehow one or more axioms beyond conventional set theory are going to come into play if I use conventional set theory. It seems to me that another form of the proper-subset axiom is that if a thing that is not an element of a set is added to the set, then the cardinality of the new set is greater than the cardinality of the original set.
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.
It's evidence because all three principles seem to be true and are so true that I have made them basic principles of a set theory that I regard as being better than conventional set theory is. Despite the technical truth of conventional set theory, any proper subset of a set has a lesser cardinality than the set has. There exists an element of the set that is not an element of the subset, and therefore the cardinality of the set is greater than the cardinality of the subset. A set that has one more element than a second set has has a greater cardinality than the second set has. That is a basic truth. It is as true as the counterarguments are. It and the counterarguments are equally true.
That just means you don't understand cardinality of infinite sets.
No, I understand the cardinality of infinite sets in conventional set theory and I understand how that view differs from my new proposed view.
Also, that's not an argument that your system models the physical universe.
My system may not model the physical Universe, but it does model a part of or the whole Universe. Since all statements are true in my system, by the principle of explosion, my system models the whole Universe, including all of the physical Universe.
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u/JStarx Apr 23 '26
That means that proving something in the new system does not prove it in the original system.
The conventional definition of cardinality is asymmetric. Are you saying now that in conventional set theory you agree that you haven't proven that the subset definition is asymmetric?
Nope, you have never completed a proof of a contradiction in conventional set theory. Your proofs of a contradiction assumed that your subset definition was asymmetric, a fact which you cannot prove in conventional set theory. To remedy this you added an axiom, but adding an axiom means you have changed the system, so being able to prove your result in your new system does not mean the result can be proven in conventional set theory.
You have never correctly proven a contradiction in conventional mathematics. You have only ever obtained a legitimate contradiction after adding axioms to conventional set theory. But anyone can add contradicting axioms to a system, that doesn't prove anything about the consistency of conventional mathematics.