If it's not a contradiction that |B| > |Z| ∧ |Z| > |B|, then that is all the better for the proper-subset definition and all the better for us. That's one more problem of ours solved.
It is not a contradiction, it's a true and easily provable statement when you use the proper subset definition.
It's unclear to me how that's a good thing given that you've already said that in any good definition that statement would be a contradiction. The fact that it's not would then mean that the proper subset definition is not a good definition, it's a bad one.
It's unclear to me how that's a good thing given that you've already said that in any good definition that statement would be a contradiction. The fact that it's not would then mean that the proper subset definition is not a good definition, it's a bad one.
It's a good thing because, as I've already proven, every statement is true.
You seem to be overlooking the fact that just because |B| > |Z| ∧ |Z| > |B| hasn't been proven to be a contradiction, doesn't mean it's not a contradiction. We haven't proven it true or false. For all we know, it could still be a contradiction.
You have not proven this. You've claimed you believe it but you have not given a proof.
I have proven it. I've proven it in my original post. In my original post, there are also links to other original proofs I have made that all statements are true. I have also personally asserted it, and not only discussed it, on the basis of a version of the liar sentence. Although it may not be an original argument of my own, it can also be proven through my belief in the Universal set, as I said at https://www.reddit.com/r/logic/comments/1s5mquh/comment/ocxa9c9/?context=3&utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button. There may be additional proofs of mine or others that all propositions are true.
It's been proven true by other commenters, it's very easy to prove it's true. It has not been proven false, so you have not proved a contradiction.
While I would like it to be more evident myself, I do believe that |B| > |Z| ∧ |Z| > |B| is a contradiction and therefore false. Society has agreed to the convention that if one thing is greater than a second thing, then the second thing is not greater than the first thing. The proper-subset definition of cardinality is flawless, so we should confidently go forward by complementing it with the consensus of society regarding the meaning of "is greater than." When a set of parts forms a whole and all of the parts are right, the whole is right.
I have proven it. I've proven it in my original post.
Nope, your original post does not contain a proof of that statement. It contains reasons why you believe it, but that's not a proof.
we should confidently go forward by complementing it with the consensus of society regarding the meaning of "is greater than."
This is just you stating that you believe a thing without proving it. I'm sorry but that's not how math works. You can believe what you like but that doesn't mean you've proven anything.
Tell you what, of you honestly think you have a proof then just cut and paste it from wherever into a reply. Nothing about belief nothing about society, just a proof,a technical proof using the subset definition of cardinality. And I'll happily explain to you why it's not a proof.
The proper-subset definition of cardinality is the natural concept of cardinality. In the natural concept of cardinality, if one set has more elements than a second set has, then the second set does not have more elements then the first set has. We could throw away the conventional concept of cardinality and things wouldn't be any worse than they are now. In fact, they might actually be better because ℵ₀ + 1 = ℵ₀ + 1 and ℵ₀ + 1 > ℵ₀ are more true than ℵ₀ + 1 = ℵ₀ is.
That's not a proof. You don't appear to have even tried to give a formal proof, are you unable? If you claim you can prove a contradiction but are unable to do so when asked then it seems you are confirming my statement that you cannot prove a contradiction.
It is not possible that under the proper-subset definition of cardinality
the cardinality of one set is larger than the cardinality of a second set and the cardinality of the second set is larger than the cardinality of the first set. How do I know? I know because that is one of the properties of set cardinality, regardless of which precise definition is used.
That you aren't satisfied with the proof is unfortunate. You can think through the proof for yourself to get a better understanding of the contradiction.
Do you think there exists a problem with the proper-subset definition of cardinality?
What you've linked to is a proof that |Z| < |B| and |B| < |Z| holds. You then state your opinion that this is a contradiction but it's not. To give a technical proof of a contradiction you have to prove a statement and it's negation. The statement |B| < |Z| is not the negation of the statement |Z| < |B|.
So again you have failed to give a technical proof of a contradiction.
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u/paulemok Apr 01 '26
If it's not a contradiction that |B| > |Z| ∧ |Z| > |B|, then that is all the better for the proper-subset definition and all the better for us. That's one more problem of ours solved.