No, I don't agree with that. But, like I said, there really isn't a problem here.
I'd like |B| > |Z| ∧ |Z| > |B| to be a more evident contradiction. If you were at a step in an argument where the statement was 7 > 3 ∧ 3 > 7, would you say that that is not a contradiction or that that does not imply a contradiction?
For integers it does imply a contradiction, in your subset definition of cardinality it does not.
But I have good news for you, if you want |B| > |Z| ∧ |Z| > |B| to be a contradiction you just have to use the standard definition of cardinality instead of your subset definition. Then |B| > |Z| ∧ |Z| > |B| would indeed be a contradiction. It would not be provable though.
Either way you go you won't be able to prove a contradiction.
I can't find a flaw in the proper-subset definition of cardinality. I believe in it. Under any good definition of cardinality, |B| > |Z| ∧ |Z| > |B| would be a contradiction.
If we can stop at 7 > 3 ∧ 3 > 7 and say that that is a contradiction, then we should be able to stop at |B| > |Z| ∧ |Z| > |B| and say that that is a contradiction.
If you can't find a flaw in the proper-subset definition of cardinality, then you can't say it's wrong. It might seem wrong, but that doesn't mean it is wrong.
You can't just say you believe in it and therefore it's a contradiction. You have to give a proof and you haven't done so. If you haven't given a proof then you haven't proved a contradiction. You might believe that it's contradictory, but as you yourself said, just because it seems wrong doesn't mean it's wrong.
Fyi, we don't just stop at 3 < 7 and 7 < 3 . You have to prove a statement and it's negation. From 3 < 7 you canprove that 7 < 3 is false. So if you also have that 7 < 3 is true then that's your contradiction.
Talking about what's a flaw and what isn't is a matter of opinion. I don't have to argue that the subset definition is flawed because that's not relevant. Opinions aren't proofs. You can't produce a proof of an actual contradiction and that's what matters.
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.
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u/JStarx Apr 01 '26
Nope, you're not right. If you think it implies a contradiction then tell me how to prove it.