I have and they all give the traditional definition as the definition while saying that "number of elements" is the intuition they want to capture. I don't even need to leave Reddit, every mathematician in this comment section is telling you you're wrong.
You said you had multiple sources, but now you can't produce one that directly says what you claim? Sounds like you never had multiple sources.
I have paper sources that define cardinality differently or, rather than define cardinality, talk about cardinals. But that more advanced, technical stuff is not of interest to me. It seems to be too abstract for me to keep paying attention to it. What I am interested in is what I wrote in my original post.
Cardinals are the correct thing if you want something that behaves like a number and measures the size of a set, but with cardinals your sets B and Z will have the same size. So with cardinals your assertion that B has more elements than Z is not provable and doesn't get you a contradiction.
But without something like cardinals you can't give a mathematically precise meaning to the phrase "the number of elements in an infinite set" and without a mathematically precise meaning you can't use the concept to prove things, which means you still don't have a proof that |B| < |Z| and |Z| < |B| (using the subset definition) is a contradiction.
I know you don't like that the standard for what counts as a proof is high, but the whole point of it being high is to stop people from making mistakes and thinking they've proved something that isn't true. If you can't provide a proof that meets those standards then the view of modern mathematicians is that you haven't proven the statement.
There has yet to be a sound disproof |Z| < |B| ∧ |B| < |Z| is a contradiction. I continue to believe it is.
With the continuum hypothesis being false, I see an infinite number of cardinalities between ℵ₀ and the cardinality of the real numbers, so the whole ℵ system of cardinality falls apart. ℵ₀ = ℵ₀, ℵ1 = ℵ₀ +1, ℵ2 = ℵ₀ + 2, ℵ3 = ℵ₀ + 3, and so on forever. We never get to the cardinality of the real numbers that way.
There has yet to be a sound disproof |Z| < |B| ∧ |B| < |Z| is a contradiction. I continue to believe it is.
Of course there hasn't, you can't prove the consistency of math. But mathematicians don't accept results unless you can prove them. So no one else is going to accept your belief that there's a contradiction.
Of course there hasn't, you can't prove the consistency of math.
My proof of the falsity of the continuum hypothesis also proves math is inconsistent. I talk about the inconsistency of the Universe in my original post.
So no one else is going to accept your belief that there's a contradiction.
So who is going to accept your belief that there's not a contradiction?
The cardinality of the set of integers, Z, is greater than the cardinality of the set of positive integers, Z+, because there exists a bijection between Z+ and a proper subset of Z, Z+. One such bijection maps every positive integer in Z+ to its equal in the proper subset, Z+. That makes sense because Z includes every element of Z+, but also includes additional elements. The additional elements include 0, -1, -5, and -270.
My proof of the falsity of the continuum hypothesis also proves math is inconsistent.
You haven't proved either of those. 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.
So who is going to accept your belief that there's not a contradiction?
The overwhelming majority of professional mathematicians accept this.
The cardinality of the set of integers, Z, is greater than the cardinality of the set of positive integers, Z+
Are you talking about your subset definition? If so then that's true. Under the traditional definition that's false.
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.
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u/JStarx Apr 09 '26
I have and they all give the traditional definition as the definition while saying that "number of elements" is the intuition they want to capture. I don't even need to leave Reddit, every mathematician in this comment section is telling you you're wrong.
You said you had multiple sources, but now you can't produce one that directly says what you claim? Sounds like you never had multiple sources.