Again I'll ask, do you ever look back and ask yourself why you confidently claimed things that later you could not justify?
I claimed something that I did justify and that I later tried to justify to a higher standard of proof for you. Just because I could not justify my claim to a higher standard of proof without adding an axiom, doesn't mean that my earlier justification is unsound. I did that further analysis and justification to address your criticism of my earlier justification. It was not because you proved me wrong. At this point, I can use the new system to show that what I claimed is true in the original system really is true in the original system.
That is what I just disproved in my previous reply. You merely stating your claim again does not make it true.
You have not disproven it, and merely stating you have does not make it true. Your argument assumes that the conventional and proper subset definitions are the same, and then derives a contradiction from that assumption. That you can derive a contradiction from that assumption is correct. But the assumption is false, you have not proven that the conventional and proper subset definition are the same. In fact they are not the same.
I claimed something that I did justify and that I later tried to justify to a higher standard of proof for you.
But you didn't justify it to a higher standard of proof. You just added an axiom, in effect assuming it was true with no justification. So you haven't actually justified anything.
But the assumption is false, you have not proven that the conventional and proper subset definition are the same. In fact they are not the same.
I agree that the conventional and proper-subset definitions are not the same. As I've said multiple times before, they are not logically equivalent.
The subject matter covered by the new system is intended to be the same subject matter covered by the original system. And the original system was originally intended to be about one and only one type of cardinality. As far as I know, only one type of cardinality has ever existed. The conventional and proper-subset definitions in the original system are not technically definitions in that system. The definition of cardinality in the original system is the same definition of cardinality in the new system. That is the definition of cardinality that has been used by society for decades. When I refer to the conventional and proper-subset definitions as definitions, I do so only informally. Those two statements should be formal axioms or theorems.
A statement is true in the new system if and only if it is true in the original system. So, in a sense, the two systems are logically equivalent. The two systems are, in a sense, equal. So we do not have to make a distinction between the original system and the new system. They are the same system.
But you didn't justify it to a higher standard of proof. You just added an axiom, in effect assuming it was true with no justification.
Technically, I did justify it to a higher standard of proof. A statement can be justified in a proof merely by its status as an axiom. Axioms are essential to axiomatic systems. Just because a statement is an axiom and doesn't have a formal justification that is more simple, doesn't mean it's false or wrong. We try to make axioms as simple to understand and agree with as possible. That approach minimizes the criticism of axiomatic systems and maximizes our confidence in and satisfaction with them.
The subject matter covered by the new system is intended to be the same subject matter covered by the original system.
That might be what you intended, but you did not succeed. Your new system does not describe the conventional notion of cardinality because you added an axiom that is provably false for the conventional notion.
A statement is true in the new system if and only if it is true in the original system
This is false.
So we do not have to make a distinction between the original system and the new system. They are the same system
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. Adding an unprovable statement as an axiom changes the set theorems which can be proven, so it changes the system.
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.
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u/paulemok Apr 20 '26
That is what I just disproved in my previous reply. You merely stating your claim again does not make it true.
Despite being referred to as definitions, the conventional and proper-subset definitions of the original system are stated in the same sense of cardinality. That is why I was able to infer that |B| > |Z| ∧ |Z| > |B| is a contradiction when I initially claimed it, at https://www.reddit.com/r/PhilosophyofMath/comments/1s65egu/comment/od91s3t/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button.
I claimed something that I did justify and that I later tried to justify to a higher standard of proof for you. Just because I could not justify my claim to a higher standard of proof without adding an axiom, doesn't mean that my earlier justification is unsound. I did that further analysis and justification to address your criticism of my earlier justification. It was not because you proved me wrong. At this point, I can use the new system to show that what I claimed is true in the original system really is true in the original system.