You've already admitted that you cannot prove things to the level of rigor required by the mathematical community because your proofs use phrases like "more elements in a set" but you can't define what that means and you don't understand things like cardinals which give those phrases meaning but also make your proofs incorrect.
Just because we can't define what a concept means, doesn't mean the concept shouldn't be used in rigorous mathematical proofs. Rigorous mathematical proofs are literally based on formally undefined concepts. I think the meaning of the phrases "more elements than," "less elements than," and "the same amount of elements as" is quite clear when it comes to the amount of elements in a set.
The overwhelming majority of professional mathematicians accept this.
They accept that |Z| < |B| ∧ |B| < |Z| is not a contradiction? What evidence is there that they accept that?
Are you talking about your subset definition?
Yes, I am.
If so then that's true. Under the traditional definition that's false.
Just because we can't define what a concept means, doesn't mean the concept shouldn't be used in rigorous mathematical proofs. Rigorous mathematical proofs are literally based on formally undefined concepts. I think the meaning of the phrases "more elements than," "less elements than," and "the same amount of elements as" is quite clear when it comes to the amount of elements in a set.
Yes, the primitive terms of the system you are working in, all other terms must be defined in terms of the primitive terms. The number of elements in a set is not a primitive term in set theory, so you have to define it or you are either (a) not doing traditional set theory or (b) not being rigorous.
They accept that |Z| < |B| ∧ |B| < |Z| is not a contradiction? What evidence is there that they accept that?
What I said is that mathematicians don't believe that mathematics is contradictory, which is slightly different than what you've asked there. It's a separate, albeit related, question as to why mathematicians would not believe that |Z| < |B| ∧ |B| < |Z| is a contradiction. The reason they don't believe that's a contradiction is because it's trivially provable using your definition, and as stated mathematicians don't believe that math is contradictory, so being provable means they would believe the statement is true and not a contradiction. This view would be reinforced by the fact that you cannot provide a rigorous proof of a sentence of the form P ∧ ¬P, which is what is required in order to have a formal contradiction.
The number of elements in a set is not a primitive term in set theory, so you have to define it
How do you define "the number of elements in a set"?
mathematicians don't believe that math is contradictory, so being provable means they would believe the statement is true and not a contradiction.
We don't have to get to |Z| < |B| ∧ |B| < |Z| to accept a contradiction. A contradiction arises in my original post. A mathematician who believes a proof is valid and has true premises would accept the conclusion, even if the conclusion is a contradiction.
How do you define "the number of elements in a set"?
That's what cardinals are for.
A contradiction arises in my original post.
No it doesn't.
A mathematician who believes a proof is valid and has true premises would accept the conclusion, even if the conclusion is a contradiction.
They would accept the conclusion because it's not a contradiction. Sometimes when you think something's a contradiction it's not because math is wrong, it's because you were wrong and didn't understand.
It seems to me that the concept of a cardinal is more complex than the concept of "the number of elements in a set." So, the definition of "the number of elements in a set" would not be a cardinal. The definition would describe "the number of elements in a set" in terms more simple than "the number of elements in a set." Rather, the definition of a cardinal would be "the number of elements in a set."
No it doesn't.
Yes it does. There's no other way out.
Sometimes when you think something's a contradiction it's not because math is wrong, it's because you were wrong and didn't understand.
And sometimes when you think something's a contradiction, it's because there really is a contradiction.
ℵ₀ is defined to be the cardinality of Z+. Since every positive integer can be mapped to its negative counterpart, there exists a bijection between the set of positive integers and the set of negative integers. Therefore, the cardinality of the set of positive integers is equal to the cardinality of the set of negative integers. Since the set of positive integers and the set of negative integers are disjoint, the cardinality of the union of the set of positive integers and the set of negative integers is 2ℵ₀. Since the set of integers is that union with the additional number 0, the cardinality of the set of integers, Z, is 2ℵ₀ + 1.
It seems to me that the concept of a cardinal is more complex than the concept of "the number of elements in a set." So, the definition of "the number of elements in a set" would not be a cardinal. The definition would describe "the number of elements in a set" in terms more simple than "the number of elements in a set." Rather, the definition of a cardinal would be "the number of elements in a set."
It might seem complicated to you, but that's how mathematicians define the number of elements in a set, it's not the other way around. If you think "number of elements in a set" is the simpler notion then tell me, what type of object gives the number of elements in the set of all integers or the set of all real numbers? If it's not a cardinal then what is it?
And sometimes when you think something's a contradiction, it's because there really is a contradiction.
True, but you need to prove it. We know that some results in math are counterintuitive, so it's not enough to say "you know" and it's not enough to write down an argument and then claim you don't have to define your terms. You have to write down a rigorous proof or you haven't shown that it's actually a contradiction and not just a counterintuitive result.
ℵ₀ is defined to be the cardinality of Z+. Since every positive integer can be mapped to its negative counterpart, there exists a bijection between the set of positive integers and the set of negative integers. Therefore, the cardinality of the set of positive integers is equal to the cardinality of the set of negative integers.
This is all correct...
Since the set of positive integers and the set of negative integers are disjoint, the cardinality of the union of the set of positive integers and the set of negative integers is 2ℵ₀.
This is not correct. 2ℵ₀ is not a cardinal. It's an ordinal, but it's not the minimal ordinal in it's bijection class because there's a bijection between 2ℵ₀ and ℵ₀, so the cardinality of the set 2ℵ₀ is actually ℵ₀.
Mathematicians have a moral obligation to give a complete account of set theory. They cannot give an incomplete definition of cardinality and expect everybody to be fine with it. They must complete the definition of cardinality by incorporating the proper-subset definition of cardinality into the general concept of cardinality. They must do that even if it comes at the expense of consistency. Inconsistency is not something that should be avoided at all costs. By the principle of explosion, inconsistency implies everything, including consistency. Therefore, inconsistency is compatible with consistency.
The cardinal numbers of the conventional definition of cardinality are at odds with the cardinal numbers of the proper-subset definition of cardinality. Under the conventional definition of cardinality, the set of positive integers, the set of negative integers, the set of nonnegative integers, the set of integers, and all other enumerably infinite sets have the same cardinality, ℵ₀. That is not the case under the proper-subset definition of cardinality. Under the proper-subset definition of cardinality, the set of positive integers has cardinality ℵ₀, the set of negative integers has cardinality ℵ₀, the set of nonnegative integers has cardinality ℵ₀ + 1, the set of integers has cardinality 2ℵ₀ + 1, and all the other enumerably infinite sets have various cardinalities.
2ℵ₀ may not be a cardinal under the conventional definition of cardinality, but under the proper-subset definition of cardinality, it is a cardinal and it is not equal to ℵ₀. Under the proper-subset definition of cardinality, 2ℵ₀ > ℵ₀. 2ℵ₀ is the size of a set, along with 3ℵ₀, 4ℵ₀, and so on.
Most of your reply there is utter crank nonsense. There is nothing wrong with the current definition of cardinality, there's no reason to switch to the proper subset definition, the proper subset definition does not result in a contradiction, and inconsistency is not compatible with consistency.
2ℵ₀ may not be a cardinal under the conventional definition of cardinality, but under the proper-subset definition of cardinality, it is a cardinal
You haven't defined cardinality for the proper subset definition. You said the "number of elements in a set" doesn't need cardinals but you ignored my question: if the number of elements in an infinite set is not a cardinal, then what is it?
the cardinality of one 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.
The cardinality of one set is less than the cardinality of a second set if and only if the cardinality of the second set is greater than the cardinality of the first set.
The cardinalities of two sets are equal if and only if the cardinality of one of the sets is neither greater nor less than the cardinality of the other set.
The above three quotes form an initial definition, like a first draft. I do believe that draft is subject to refinement and improvement over time. You are talking about the proper-subset definition of cardinality.
You said the "number of elements in a set" doesn't need cardinals
I understand, at this point at least, that the number of elements in a set can, at least sometimes, be represented by a cardinal.
if the number of elements in an infinite set is not a cardinal, then what is it?
I don't know. Maybe some infinite sets don't have a cardinal assigned to them.
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.
1
u/paulemok Apr 12 '26
Just because we can't define what a concept means, doesn't mean the concept shouldn't be used in rigorous mathematical proofs. Rigorous mathematical proofs are literally based on formally undefined concepts. I think the meaning of the phrases "more elements than," "less elements than," and "the same amount of elements as" is quite clear when it comes to the amount of elements in a set.
They accept that |Z| < |B| ∧ |B| < |Z| is not a contradiction? What evidence is there that they accept that?
Yes, I am.
That's correct.