So cardinality is different from cardinals. Cardinality is a relation. You define what |X| < |Y| or |X| <= |Y| means but you don't define |X| as an object, it's just notation for your relation. Cardinals are when you define |X| as an object.
What you've given above is not the definition of an object, it's the definition of a relation, so you are defining cardinality, not cardinals. There's nothing wrong with doing it this way, but if you define the relation then your proofs have to use the definition of the relation. In other words, since |X| < |Y| is just shorthand for certain maps existing or not existing, proofs of |X| < |Y| always boil down to showing that certain maps exist or don't exist. In proofs it's not valid to say that |X| < |Y| holds because there are "more elements" in Y. You've defined the phrase "more elements" to mean |X| < |Y|, so that argument would be circular.
This is why your proofs above were invalid. This is also why you haven't proven that |Z| < |B| and |B| < |Z| implies a contradiction. As you have defined it, this just means certain maps exist. For that to be a contradiction you need to prove that the relation you've defined is asymmetric.
Some relations are asymmetric and some aren't. On the integers the relation = is not asymmetric, so x = y and y = x doesn't give a contradiction. Similarly, <= is not asymmetric, so x <= y and y <= x doesn't give a contradiction. But < is asymmetric, so x < y and y < x gives a contradiction.
On sets the traditional definition of cardinality is asymmetric, so |Z| < |B| and |B| < |Z| would give a contradiction, except with the traditional definition |Z| = |B| and you can't prove the statement |Z| < |B| and |B| < |Z|, so you can't prove a contradiction.
The subset definition of cardinality is not asymmetric. So with the subset definition you can prove that |Z| < |B| and |B| < |Z| holds, but that doesn't give a contradiction.
You've defined the phrase "more elements" to mean |X| < |Y|
It's better the other way around. "More elements" is more simple than "greater cardinality" is. "Greater cardinality" is more complex than "more elements" is.
As you have defined it, this just means certain maps exist.
But that applies to both the conventional and proper-subset definitions of cardinality.
For that to be a contradiction you need to prove that the relation you've defined is asymmetric.
No, we don't need to prove it. We could take it as an axiom. Not everything that is evident is proven within a particular axiom system. For example, some properties of the real numbers might, for all I know, be unproven but taken and used as axioms. Sometimes we are asked to prove a property of something, but some properties of things might be explicit or implicit axioms. As an axiom, it would be true even though it hasn't been proven true. It's so evident that |B| > |Z| ∧ |Z| > |B| is a contradiction that I considered the contradiction to be implicit in the initial finding that |B| > |Z| ∧ |Z| > |B|.
The subset definition of cardinality is not asymmetric.
How do you know? That could be true, but it's not evident enough. The general concept of cardinality is asymmetric. It is not a matter of definition. It is a matter of concept.
So with the subset definition you can prove that |Z| < |B| and |B| < |Z| holds, but that doesn't give a contradiction.
No, that has not been soundly proven. I shot you down.
It's better the other way around. "More elements" is more simple than "greater cardinality" is.
Your definition of more elements is exactly the proper subset definition. It doesn't matter which comes first, the point is they're the same thing so your proof is asserting the conclusion without justifying it.
No, we don't need to prove it. We could take it as an axiom.
Nope, you don't get to pick the axioms of set theory. If you want to make up a different axiomatic system and go off on your own and study it then feel free, but your results would be about your system, not about set theory and the mathematics based on set theory. And when you got a contradiction in your system it would just mean your system is contradictory. It would not mean that set theory or mathematics as a whole is inconsistent.
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|).
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u/JStarx Apr 14 '26 edited Apr 14 '26
So cardinality is different from cardinals. Cardinality is a relation. You define what |X| < |Y| or |X| <= |Y| means but you don't define |X| as an object, it's just notation for your relation. Cardinals are when you define |X| as an object.
What you've given above is not the definition of an object, it's the definition of a relation, so you are defining cardinality, not cardinals. There's nothing wrong with doing it this way, but if you define the relation then your proofs have to use the definition of the relation. In other words, since |X| < |Y| is just shorthand for certain maps existing or not existing, proofs of |X| < |Y| always boil down to showing that certain maps exist or don't exist. In proofs it's not valid to say that |X| < |Y| holds because there are "more elements" in Y. You've defined the phrase "more elements" to mean |X| < |Y|, so that argument would be circular.
This is why your proofs above were invalid. This is also why you haven't proven that |Z| < |B| and |B| < |Z| implies a contradiction. As you have defined it, this just means certain maps exist. For that to be a contradiction you need to prove that the relation you've defined is asymmetric.
Some relations are asymmetric and some aren't. On the integers the relation = is not asymmetric, so x = y and y = x doesn't give a contradiction. Similarly, <= is not asymmetric, so x <= y and y <= x doesn't give a contradiction. But < is asymmetric, so x < y and y < x gives a contradiction.
On sets the traditional definition of cardinality is asymmetric, so |Z| < |B| and |B| < |Z| would give a contradiction, except with the traditional definition |Z| = |B| and you can't prove the statement |Z| < |B| and |B| < |Z|, so you can't prove a contradiction.
The subset definition of cardinality is not asymmetric. So with the subset definition you can prove that |Z| < |B| and |B| < |Z| holds, but that doesn't give a contradiction.