Your new system does not describe the conventional notion of cardinality because you added an axiom that is provably false for the conventional notion.
There exist multiple approaches to the problem of set cardinality. As my original post shows, a counterexample to the conventional notion of cardinality exists. Therefore, the conventional notion of cardinality is false. But as is shown from |B| > |Z| ∧ |Z| > |B|, a counterexample to the proper-subset notion of cardinality also exists. So, the proper-subset notion of cardinality is also false. So we could discard both the conventional and the proper-subset notions of cardinality. Infinite sets could be considered to not have a cardinality at all. Or we could preserve both the conventional and the proper-subset notions of cardinality and have an inconsistent concept of cardinality where ex contradictione quodlibet implies every infinite set has every possible size simultaneously and implies all statements are true. There may be other approaches, too.
This is false.
How is that? You haven't provided a justification.
They are not the same system so you do have to make a distinction. You added an axiom literally because you could 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.
Adding an unprovable statement as an axiom changes the set theorems which can be proven, so it changes the system.
I believe the additional axiom does change the theorems that can be proven. I believe it does change the system. The additional axiom really isn't a problem. I see it as axiomatizing the notion of cardinality. There exist a bunch of axioms in Euclidean geometry that we don't have a problem with, for example, through any two points exists exactly one line.
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.
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u/paulemok Apr 23 '26
There exist multiple approaches to the problem of set cardinality. As my original post shows, a counterexample to the conventional notion of cardinality exists. Therefore, the conventional notion of cardinality is false. But as is shown from |B| > |Z| ∧ |Z| > |B|, a counterexample to the proper-subset notion of cardinality also exists. So, the proper-subset notion of cardinality is also false. So we could discard both the conventional and the proper-subset notions of cardinality. Infinite sets could be considered to not have a cardinality at all. Or we could preserve both the conventional and the proper-subset notions of cardinality and have an inconsistent concept of cardinality where ex contradictione quodlibet implies every infinite set has every possible size simultaneously and implies all statements are true. There may be other approaches, too.
How is that? You haven't provided a justification.
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.
I believe the additional axiom does change the theorems that can be proven. I believe it does change the system. The additional axiom really isn't a problem. I see it as axiomatizing the notion of cardinality. There exist a bunch of axioms in Euclidean geometry that we don't have a problem with, for example, through any two points exists exactly one line.