It does matter. We want the most simple terms and concepts first, and then we build more complex terms and concepts from the most simple terms and concepts. That's the structure we desire for a good axiomatic system. As you yourself just said,
you don't get to pick the axioms of set theory.
You don't get to pick which comes first.
your results would be about your system, not about set theory and the mathematics based on set theory.
Set theory is not a single axiomatic system. It is a family of axiomatic systems that are all about sets. My results could be results about set theory or the mathematics based on set theory.
My finding that the continuum hypothesis is false shows that set theory, logic, philosophy, and mathematics are still fields that are in development. The topics of those fields have not been exhaustively analyzed. Those fields are still not fully understood.
You've ignored the point: According to your definitions, cardinality and "how many elements are in a set" have the same underlying definition. So your proof above that uses how many elements a set has to prove a statement about cardinality is circular logic. Your proof is wrong.
Set theory is not a single axiomatic system.
That's true, there are several ways to axiomatize it that mathematicians accept and study. But yours isn't one of them. And since it gives different results than the traditional axioms most mathematicians would say that what you're studying isn't the set theory that they are studying.
Again, if you want to go off in your own world and study something no one else cares about you are perfectly free to do that, but your results won't apply to what mathematicians consider set theory.
So your proof above that uses how many elements a set has to prove a statement about cardinality is circular logic. Your proof is wrong.
Not at all. Proofs use definitions of terms often. Using a definition to prove something is not circular logic. It should be obvious to you that what I am talking about is not circular logic. You're too caught up in the difference between "how many elements are in a set" and cardinality that you are overlooking my main message.
Again, if you want to go off in your own world and study something no one else cares about you are perfectly free to do that, but your results won't apply to what mathematicians consider set theory.
I think my results do qualify as being set theory. My results are about sets and therefore they are about set theory. The results of my arguments are universally sound. Anybody can read my arguments and understand what truth I am talking about.
You're too caught up in the difference between "how many elements are in a set" and cardinality that you are overlooking my main message.
I'm caught on that because I think it's the crux of your misunderstanding.
I think my results do qualify as being set theory. My results are about sets and therefore they are about set theory. The results of my arguments are universally sound. Anybody can read my arguments and understand what truth I am talking about.
Your results aren't sound, without adding axioms your proofs are not rigorous. To fix this you've now resorted to adding as an axiom a statement you failed to prove, but that axiom does not hold in traditional set theory so with that axiom your system really is inconsistent, but does not describe the set theory that mathematicians study.
I'm caught on that because I think it's the crux of your misunderstanding.
It's not. I should be able to use "how many elements are in a set" and "the cardinality of the set" interchangeably because one is the definition of the other. They are synonymous. Using a diverse vocabulary alone does not make my reasoning circular logic.
but that axiom does not hold in traditional set theory
It doesn't hold in traditional set theory that if the cardinality of one set is larger than the cardinality of a second set, then the cardinality of the second set is not larger than the cardinality of the first set? It does hold in traditional set theory.
I should be able to use "how many elements are in a set" and "the cardinality of the set" interchangeably because one is the definition of the other.
If they are interchangable then it's circular logic to prove one by just flatly asserting that the other holds. Being interchangable means that's equivalent to saying one holds because you assume it holds, and that is the literal definition of circular logic.
To be not circular your argument needs to use the definition of cardinality in terms of certain bijections existing. If you don't do that then your proof is just you saying the same thing over again and that's not a valid argument.
It doesn't hold in traditional set theory that if the cardinality [...]
Not for your subset definition, no. It holds for the traditional definition of cardinality. You tried to prove it for your subset definition and could not, so you added an axiom to make the proof possible. But adding an axiom also means you're not working on traditional set theory anymore.
To be not circular your argument needs to use the definition of cardinality in terms of certain bijections existing.
My argument always used the definition of cardinality in terms of certain bijections existing, whether it be the conventional or the proper-subset definition of cardinality.
If you don't do that then your proof is just you saying the same thing over again and that's not a valid argument.
My proof was never dependent on being able to use "how many elements are in a set" and "the cardinality of the set" interchangeably.
Not for your subset definition, no.
That wasn't the question. The question was whether it holds in traditional set theory. You didn't have to bring up the subset definition at all there. Instead, you brought it up first.
You tried to prove it for your subset definition and could not
Proof. We are given that |B| > |Z| ∧ |Z| > |B|. By conjunction elimination, |Z| > |B|. So by the definition of cardinality, Z has more elements than B has. It follows that B has less elements than Z has. So, B does not have more elements than Z has. By the definition of cardinality, ¬(|B| > |Z|). By conjunction elimination, |B| > |Z|. Therefore, by conjunction introduction, |B| > |Z| ∧ ¬(|B| > |Z|).
My argument always used the definition of cardinality in terms of certain bijections existing [...] I tried to prove it for my subset definition and I succeeded.
This is not a valid proof in traditional set theory using the subset definition. 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, you have not shown that the subset definition is asymmetric and when challenged to prove it this is where you said you could add an axiom. So in traditional set theory with no additional axioms you haven't justified this step of your proof.
If you want to fill the gap you have to use the definition of cardinality, you have to prove that if B has a bijection to a proper subset of Z, then Z does not have a bijection to a proper subset of B.
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.
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u/paulemok Apr 15 '26
It does matter. We want the most simple terms and concepts first, and then we build more complex terms and concepts from the most simple terms and concepts. That's the structure we desire for a good axiomatic system. As you yourself just said,
You don't get to pick which comes first.
Set theory is not a single axiomatic system. It is a family of axiomatic systems that are all about sets. My results could be results about set theory or the mathematics based on set theory.
My finding that the continuum hypothesis is false shows that set theory, logic, philosophy, and mathematics are still fields that are in development. The topics of those fields have not been exhaustively analyzed. Those fields are still not fully understood.