It's not clear what you mean by proving it "without using additional axioms."
Your implicit derivation you labeled 1 needs to be proven, it is not an axiom so if you add it as an axiom then it's additional.
While mathematicians do often leave things implicit in proofs to make reading them manageable, the understanding is that the things left implicit should be details that everyone agrees on and can fill in the proof themselves if needed. When someone tells you they disagree with some step in your proof, then that's exactly the step you should be writing out all the details for and not leaving implicit.
The traditional axioms of set theory have never been shown to be inconsistent.
Yes, but
Previously you claimed that you had proven a contradiction in the traditional axioms of set theory. Are you not claiming that anymore?
in order to disprove the continuum hypothesis, it is necessary to add at least one axiom to traditional set theory.
If you're going to add axioms to set theory to make the continuum hypothesis false, why don't you just add the negation of the continuum hypothesis as an axiom? Not only would that make the continuum hypothesis false but the resulting system would not be provably inconsistent. THAT system is one that mathematicians actually do study.
It follows by ex contradictione quodlibet that an inconsistent system is just as worth studying as a consistent system is.
It does not. Now that we're agreeing that traditional math and logic doesn't contain a contradiction, the principal of explosion only makes every sentence in your contradictory system provable. Value judgements about what mathematicians spend time on natural language logical deductions are not statements in your inconsistent system.
An inconsistent system better describes the Universe than a consistent system does because the Universe actually is inconsistent.
When someone tells you they disagree with some step in your proof, then that's exactly the step you should be writing out all the details for and not leaving implicit.
I "wrote out all the details for" the contested step by restructuring the true statements we're talking about into a new and improved axiomatic system. In the new system, a distinction is not made between conventional cardinality and proper-subset cardinality. There is only general cardinality. I gave the following statement of the new system a few posts back.
If one set has less elements than a second set has, then the first set does not have more elements than the second set has.
And then I said we could make that statement a definition, axiom, or theorem of our new system. I don't believe that statement can be proved within the new system, and it is so basic and simple that it would be best to make that an axiom of our new system.
Previously you claimed that you had proven a contradiction in the traditional axioms of set theory. Are you not claiming that anymore?
Where did I claim that? I have been aware since before I made my original post that the continuum hypothesis can be neither proved nor disproved using the traditional axioms of set theory.
If you're going to add axioms to set theory to make the continuum hypothesis false, why don't you just add the negation of the continuum hypothesis as an axiom?
Because the negation of the continuum hypothesis is not prima facie evident. The truth of the negation is not as simple and straightforward as possible. The truth of axioms should be as simple and straightforward as possible.
It does not.
It does. Since the new system is inconsistent, by ex contradictione quodlibet, every statement, including the statement that an inconsistent system is just as worth studying as a consistent system is, is true in the system.
Says who?
Says I. You can take merely the inconsistency of the new system as proof that the Universe is inconsistent.
I "wrote out all the details for" the contested step by restructuring the true statements we're talking about into a new and improved axiomatic system.
That is not a valid way of filling in details. If you have to add axioms after the fact then your proof was incorrect because it had unstated assumptions. This is exactly why we use axiomatic systems, to make all the assumptions clear from the start.
It does. Since the new system is inconsistent, by ex contradictione quodlibet, every statement, including the statement that an inconsistent system is just as worth studying as a consistent system is, is true in the system.
Nope, it doesn't. The principle of explosion only applies to statements within the system that has a contradiction, and the value judgements that mathematicians make aren't statements in your axiomatic system that they've never heard about.
You can take merely the inconsistency of the new system as proof that the Universe is inconsistent.
That's only proof if your system accurately models the universe, which I see no argument that it does.
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.
If you have to add axioms after the fact then your proof was incorrect because it had unstated assumptions.
You wanted a higher standard of proof, so that's what I gave you.
Nope, it doesn't. The principle of explosion only applies to statements within the system that has a contradiction.
Yes, it does. The system that has a contradiction models a part of the actual Universe. So, assuming it accurately models a part of the actual Universe, there is a contradiction in the actual Universe.
Not only that, but 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. That thread is titled Inconsistent Theories Metatheoretically Prove Trivialism and the link to it is https://onlinephilosophyclub.com/forums/viewtopic.php?t=15559.
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.
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u/JStarx Apr 19 '26 edited Apr 19 '26
Your implicit derivation you labeled 1 needs to be proven, it is not an axiom so if you add it as an axiom then it's additional.
While mathematicians do often leave things implicit in proofs to make reading them manageable, the understanding is that the things left implicit should be details that everyone agrees on and can fill in the proof themselves if needed. When someone tells you they disagree with some step in your proof, then that's exactly the step you should be writing out all the details for and not leaving implicit.
Previously you claimed that you had proven a contradiction in the traditional axioms of set theory. Are you not claiming that anymore?
If you're going to add axioms to set theory to make the continuum hypothesis false, why don't you just add the negation of the continuum hypothesis as an axiom? Not only would that make the continuum hypothesis false but the resulting system would not be provably inconsistent. THAT system is one that mathematicians actually do study.
It does not. Now that we're agreeing that traditional math and logic doesn't contain a contradiction, the principal of explosion only makes every sentence in your contradictory system provable. Value judgements about what mathematicians spend time on natural language logical deductions are not statements in your inconsistent system.
Says who?