I guess I was not clear enough. In your original post, you wrote “On one hand, there are countably many definitions …” and “On the other hand, Cantor’s diagonal argument applies here, too. …”. So, you talked about two statements—“On one hand, (1)”, “On the other hand, (2)”. I would expect that when someone says “One one hand, …, but on the other hand, …”, what they say in those ellipses should contradict each other. So, in my previous comment, I just wanted to point out that (2) does not contradict (1) because countable infinity + 1 is still countable infinity.
take all the iterations you need, even infinitely many of them
Could you clarify how I would construct that?
For example, what is the “next cardinality” after countable?
I didn’t say “the next cardinality”. I said “a higher cardinality”.
Cantor’s diagonal argument is not “I can find +1, and n+1 is more than n”, which indeed would be wrong. It is “if you believe that you have a countable set that already contains all of them, I can still find +1 it does not contain”. The problem is not that +1 is more, but that there is a contradiction between the assumption that you have the things enumerated, and the fact that you have not—because there is at least one (but probably much more) item outside the enumeration.
I am sorry, this is getting complicated and my free time budget is short these days, so… I’m “tapping out”.
I guess I was not clear enough. In your original post, you wrote “On one hand, there are countably many definitions …” and “On the other hand, Cantor’s diagonal argument applies here, too. …”. So, you talked about two statements—“On one hand, (1)”, “On the other hand, (2)”. I would expect that when someone says “One one hand, …, but on the other hand, …”, what they say in those ellipses should contradict each other. So, in my previous comment, I just wanted to point out that (2) does not contradict (1) because countable infinity + 1 is still countable infinity.
Could you clarify how I would construct that?
I didn’t say “the next cardinality”. I said “a higher cardinality”.
Cantor’s diagonal argument is not “I can find +1, and n+1 is more than n”, which indeed would be wrong. It is “if you believe that you have a countable set that already contains all of them, I can still find +1 it does not contain”. The problem is not that +1 is more, but that there is a contradiction between the assumption that you have the things enumerated, and the fact that you have not—because there is at least one (but probably much more) item outside the enumeration.
I am sorry, this is getting complicated and my free time budget is short these days, so… I’m “tapping out”.