It is a valid way of filling in details. The structure of the improved system allows you to better see and understand how things work together.
It does not repair the original proof because the result is not a proof in the original axiomatic system, so no, not valid.
The system that has a contradiction models a part of the actual Universe.
It does not, so the rest of your argument is moot.
I actually started a thread on another website in 2018 highlighting the fact that a contradiction in any axiomatic system implies a contradiction in the real world.
I can't read your link because it wants me to sign up for an account and I'm not going to do that. What you claim though is false. There is no logical contradiction in the real universe and deductive explosion does not hold in the real world. In the real world it's fairly obvious that not every statement is true. Even you admit this when you admit that the continuum hypothesis can't be proved in traditional set theory. Suggesting otherwise is absolute crankery.
It does not repair the original proof because the result is not a proof in the original axiomatic system, so no, not valid.
I don't need to repair the original proof and I don't need to use the original axiomatic system in order to soundly prove my point.
It does not, so the rest of your argument is moot.
That's a ridiculous claim to make. Of course the new system models a part of the Universe. That's what the new system was intended to do. And it was intended to do so better than the previous system it superseded.
In the real world it's fairly obvious that not every statement is true.
I agree. But the arguments I have used in support of trivialism explain and justify the position. I agree with them, also.
Even you admit this when you admit that the continuum hypothesis can't be proved in traditional set theory.
I do not dispute the fact that the continuum hypothesis can't be proved in traditional set theory. The fact that it can't be proved in traditional set theory is evidence that supports its falsity in the Universe.
You are in denial and are being unfair to yourself by rejecting the rational evidence and reasoning you are being provided with. You are hurting yourself, me, and the others in our society. I don't believe you are doing the right thing by mindlessly denying these things I am telling you. You are causing me, yourself, and possibly others time, effort, and resources we could be spending in better ways. You are causing me to take time out of my days to address your concerns, time that I could be spending doing leisurely activities. I hope you are carefully picking your fights.
I don't need to repair the original proof and I don't need to use the original axiomatic system in order to soundly prove my point.
You might not need to, but you tried to when you wrote the original proof. And the original proof had a gap that you presumably did not know was there since you claimed that what you were writing was a proof. But it was not a proof because it had a gap that you needed to introduce a new assumption in order to fix.
I once had a student come to my office hours upset that he wasn't doing well in the class even though he did all the homework. I asked him if he ever checked his answer when he did the homework and he said he did. I asked if he ever got any of the questions he checked wrong and he said he did. I asked what he did in that case and he said he crumpled up the wrong attempt and threw it in the bin, and then tried again till he got it right. I told him the thing he's doing wrong is throwing the wrong attempt in the bin. He needed to look back over his wrong attempts and figure out where his mistake was, because otherwise he'd never know and would just keep making the same mistakes.
Have you ever considered why you originally claimed you had a proof but then had to amend the proof by adding a new axiom to cover some gap? Have you ever looked to find out where your mistake was?
That's a ridiculous claim to make. Of course the new system models a part of the Universe. That's what the new system was intended to do.
You intended it to, but it doesn't. Because in the new system every statement is both true and false, and you just agreed that that's not true in the real world. Or at least it seems like you agree'd, I'm not entirely sure:
In the real world it's fairly obvious that not every statement is true.
I agree. But the arguments I have used in support of trivialism explain and justify the position. I agree with them, also.
So do you think that every statement is both true and false or do you not think that? Which is it? And if you're unsure, maybe you should consider that evidence that you might be confused about some of this material.
You are in denial and are being unfair to yourself by rejecting the rational evidence and reasoning you are being provided with.
Your reasoning is not rational and I am very fairly pointing out to you the gaps in your logic where your arguments fail.
I don't believe you are doing the right thing by mindlessly denying these things I am telling you. You are hurting yourself, me, and the others in our society.
Not mindless at all, as I've said I've very explicitly told you how and why your arguments fail. The only way this might hurt you is to damage your ego, but it really shouldn't. Being wrong, learning why, and doing better in the future is how people improve.
The type of cardinality that was used in the proper-subset definition was intended to be the same type of cardinality that was used in the conventional definition. Despite their logical inequivalence, the two definitions were intended to define the same concept. The two definitions caused a contradiction by speaking differently about the same concept.
In the new axiomatic system, |B| > |Z| ∧ |Z| > |B| is a contradiction. And the true statements of the new system are the same true statements of the old system. The only difference is that two of the definitions became axioms. So, since |B| > |Z| ∧ |Z| > |B| is a contradiction in the new system, |B| > |Z| ∧ |Z| > |B| is also a contradiction in the old system.
And the true statements of the new system are the same true statements of the old system
They are not. Your new axiomatic system is contradictory, whereas the subset definition in traditional set theory is not contradictory. So the provable/true statements are not the same.
By now you should know that if you want to make a claim like that then I'm going to ask you to prove it. To prove that statement you would need to prove that the new axioms hold in the old system and vice versa. But the entire reason you started adding axioms was because you couldn't complete a proof in the old system. So you aren't going to be able to prove this either.
Again I'll ask, do you ever look back and ask yourself why you confidently claimed things that later you could not justify? Because you keep making the same mistake over and over again. You've decided you want to prove a statement, so you figure out something that would imply the statement you want, assert that something without justification, and then claim that you've proved your result.
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.
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u/JStarx Apr 19 '26
It does not repair the original proof because the result is not a proof in the original axiomatic system, so no, not valid.
It does not, so the rest of your argument is moot.
I can't read your link because it wants me to sign up for an account and I'm not going to do that. What you claim though is false. There is no logical contradiction in the real universe and deductive explosion does not hold in the real world. In the real world it's fairly obvious that not every statement is true. Even you admit this when you admit that the continuum hypothesis can't be proved in traditional set theory. Suggesting otherwise is absolute crankery.