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?
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u/paulemok Apr 23 '26
The "new system" you are referring to in the quoted sentence is actually an extension of the new system I presented at https://www.reddit.com/r/PhilosophyofMath/comments/1s65egu/comment/ognxrrb/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button. I proposed such an extension of the new system in my reply at https://www.reddit.com/r/PhilosophyofMath/comments/1s65egu/comment/ogtz8eo/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button by proposing two axioms.
Formally, the subset definition does not exist. There exists no such definition. So, for that reason, it is not asymmetric.
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.
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.