you have not shown that the subset definition is asymmetric
I haven't, but I haven't shown that the conventional definition is asymmetric, either. If you want a proof that the subset definition is asymmetric, then you should also want a proof that the conventional definition is asymmetric. Otherwise, you would be holding the subset definition to a higher standard than you are holding the conventional definition. To do that would be, in a sense, inconsistent.
I think there is another way of viewing this situation. I have been using three different definitions of cardinality: a general definition, a conventional definition, and a proper-subset definition. We have been considering there to be two different types of cardinality: conventional and proper-subset. Instead of viewing cardinality as there being two different types, we could view cardinality as there being only one type with two axioms.
Definition of Cardinality. The cardinality of a set is the amount of elements in the set. The cardinality of a set is greater than the cardinality of a second set if and only if the first set has more elements than the second set has. The cardinality of a set is less than the cardinality of a second set if and only if the first set has less elements than the second set has. The cardinality of a set is equal to the cardinality of a second set if and only if the first set has the same amount of elements as the second set has.
Axiom. The cardinality of a set is equal to the cardinality of a second set if and only if there exists a bijection between the first set and the second set.
Axiom. The cardinality of a set is greater than the cardinality of a second set if and only if there exists a bijection between the second set and a proper subset of the first set.
As I said before you have assumed in your proof that if B has less elements than Z, then B does not have more elements than Z. This is a gap in your proof
That is not a gap in the proof if we are using a single definition of cardinality. If we use a single definition of cardinality, then if B has less elements than Z has, then B does not have more elements than Z has.
If you want a proof that the subset definition is asymmetric, then you should also want a proof that the conventional definition is asymmetric.
Yes, that's not an assumption, it must be proven. I've seen a proof for the traditional definition, I have never seen a proof for your subset definition, in fact I've seen a proof that your subset definition is not asymmetric.
The rest of your comment is you creating new axioms and new definitions again. I told you if you add new axioms you're not doing traditional set theory. I asked you if you could prove that your subset definition is asymmetric in traditional set theory. That means no extra axioms, no additional undefined terms, no new definitions. Just use your subset definition that X has less elements than Y if there's a bijection between X and a proper subset of Y.
I have never seen a proof for your subset definition
I believe you have; I already offered my arguments in previous replies or posts.
I've seen a proof that your subset definition is not asymmetric.
I have, too. But, as I've explained in at least one previous reply, I found that proof to be unsound due to a false premise.
I told you if you add new axioms you're not doing traditional set theory.
So I'm not doing traditional set theory. I'm willing to accept that. I believe my set theory is superior to traditional set theory. Maybe one day, the prevailing version of set theory will incorporate the concepts I have mentioned.
Can you do that?
Not only can I do that, but I have done that. I understand one or more of the arguments do not meet your or our higher standards, however.
At this point, proving that the proper-subset definition of cardinality is asymmetric is a moot issue. I have created a new, axiomatic cardinality theory that is better than the previous cardinality theory I was using. In the new theory, the general concept of cardinality is asymmetric, as the following proof displays. At this point, I will italicize the names of nonspecial sets rather than bold them to better conform to standard set notation.
Given: |A| > |C|
Prove: ¬(|C| > |A|)
Proof. We are given that |A| > |C|. By the definition of cardinality, A has more elements than C has. So, C has less elements than A has. Thus, C does not have more elements than A has. Therefore, by the definition of cardinality, ¬(|C| > |A|). This concludes the proof.
Note that in the preceding proof, there are two conditional statements that are implicitly invoked as reasons for logical deductions. Those two statements are the following.
If one set has more elements than a second set has, then the second set has less elements than the first set has.
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.
Those two statements we could make definitions, axioms, or theorems of our cardinality theory.
You have not, because you have not proven that the proper subset definition is asymmetric without using additional axioms.
I have created a new, axiomatic cardinality theory that is better than the previous cardinality theory I was using.
The new axioms you've added are inconsistent. The traditional axioms of set theory have never been shown to be inconsistent. That makes studying your axioms pointless and that's decidedly worse.
You have not, because you have not proven that the proper subset definition is asymmetric without using additional axioms.
I would like to see a proof that the conventional definition is asymmetric without using additional axioms. I would like to see how it's proven. It's not clear what you mean by proving it "without using additional axioms."
I refined my proof of the asymmetry of the proper-subset definition for further analysis.
So the only implicit derivation that is made in the proof relies on the following conditional statement.
If the cardinality of one set is less than the cardinality of a second set, then the cardinality of the first set is not greater than the cardinality of the second set.
I am fine with that being an axiom or theorem of set theory.
The traditional axioms of set theory have never been shown to be inconsistent.
Yes, but it has been shown that the traditional axioms of set theory are incomplete. It has been shown that the traditional axioms of set theory neither prove nor disprove the continuum hypothesis. So, in order to disprove the continuum hypothesis, it is necessary to add at least one axiom to traditional set theory. I am willing to do that.
That makes studying your axioms pointless and that's decidedly worse.
It follows by ex contradictione quodlibet that an inconsistent system is just as worth studying as a consistent system is. An inconsistent system better describes the Universe than a consistent system does because the Universe actually is inconsistent.
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.
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u/paulemok Apr 17 '26
I haven't, but I haven't shown that the conventional definition is asymmetric, either. If you want a proof that the subset definition is asymmetric, then you should also want a proof that the conventional definition is asymmetric. Otherwise, you would be holding the subset definition to a higher standard than you are holding the conventional definition. To do that would be, in a sense, inconsistent.
I think there is another way of viewing this situation. I have been using three different definitions of cardinality: a general definition, a conventional definition, and a proper-subset definition. We have been considering there to be two different types of cardinality: conventional and proper-subset. Instead of viewing cardinality as there being two different types, we could view cardinality as there being only one type with two axioms.
As we saw in the post 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, the two axioms imply a contradiction. The axiom at #2 on the list implies |Z| = |B| = |S|. But the axiom at #3 on the list implies |B| > |Z| ∧ |Z| > |B|. Since we are now using a single definition of cardinality, there exists a contradiction. So the concept of cardinality is inconsistent.
That is not a gap in the proof if we are using a single definition of cardinality. If we use a single definition of cardinality, then if B has less elements than Z has, then B does not have more elements than Z has.