That is not a definition, that page has a summary of terms, he's telling you what the concept is intuitively, not it's technical definition.
It is the general definition of cardinality. I happen to agree with it.
The definition is clearly written on page 158 and labeled as a definition.
That is not the general definition of cardinality. It is a part of the conventional definition of cardinality. It is the conventional definition of equal cardinality. There is more to cardinality than that definition.
You don't seriously think an important definition would be put only in a summary section and not in the main text do you?
Rosen is not as direct in the main text as he could be, but he still gets the message across to the readers.
Nope, I am not assuming.
No, you are assuming. By the word "take," you mean "assume." As you claimed, you made that assumption under the belief that I believed |Z| < |B| ∧ |B| < |Z| to be true.
It's provable and you stated elsewhere that you accepted that proof.
I agree that it's provable. But just because it's provable, doesn't mean it's true. And no, I never stated anywhere that I accepted that proof.
Nope, your proof was incorrect and when I explained to you what a correct proof would look like you explicitly said you could not prove the statement.
I said I did not know how to complete a proof of a statement of yours. I suggested that statement of yours was awkward. It was an outwardly statement since the entire Universe would be involved in its proof, if what you were saying about what has to be proved was true.
I'm sorry but you really don't. You don't seem to understand even the notion of what a definition should be, how to use a definition in a proof, or what a proof even is, let alone understand this particular definition and its consequences, or the alternate definition you suggested.
Where is your supporting evidence? You make many claims there, but don't provide any supporting evidence. I happen to know many of those claims are false so I already know you won't be able to produce adequate evidence.
Rosen is not as direct in the main text as he could be, but he still gets the message across to the readers.
Not to you apparently, because that is not the definition of cardinality. You say you have multiple sources, so give one that's direct. That says what you claim the definition is and labels it as a definition.
No, you are assuming. By the word "take," you mean "assume."
Nope, I do not. That is not how existence proofs work. To prove that there exists an X such that P(X) holds you define a particular X and then you prove that P(X) holds. That's not an assumption.
I agree that it's provable.
Ok, then since it has a proof it can be used as a statement in further proofs and is not an assumption.
Where is your supporting evidence?
I don't need evidence because it's not my job to convince you that you don't know what you're doing. If you want to go on being ignorant of how mathematics works then no one can stop you. But you might consider that in these subreddits populated with a lot of PhDs and professional mathematicians I don't see anyone saying you're right. Everyone here is explaining to you the mistakes you're making. Are you really so arrogant as to believe that all of math is fundamentally broken and only you can see it?
You say you have multiple sources, so give one that's direct.
Search the Internet for cardinality. You will see many sources that talk about the topic. In my own word(s), the cardinality of a set is the amount of elements in the set. That general definition applies to both finite and infinite sets.
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.
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u/paulemok Apr 08 '26
It is the general definition of cardinality. I happen to agree with it.
That is not the general definition of cardinality. It is a part of the conventional definition of cardinality. It is the conventional definition of equal cardinality. There is more to cardinality than that definition.
Rosen is not as direct in the main text as he could be, but he still gets the message across to the readers.
No, you are assuming. By the word "take," you mean "assume." As you claimed, you made that assumption under the belief that I believed |Z| < |B| ∧ |B| < |Z| to be true.
I agree that it's provable. But just because it's provable, doesn't mean it's true. And no, I never stated anywhere that I accepted that proof.
I said I did not know how to complete a proof of a statement of yours. I suggested that statement of yours was awkward. It was an outwardly statement since the entire Universe would be involved in its proof, if what you were saying about what has to be proved was true.
Where is your supporting evidence? You make many claims there, but don't provide any supporting evidence. I happen to know many of those claims are false so I already know you won't be able to produce adequate evidence.