this post was submitted on 07 May 2024
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countable infinite set are unique up-to bijection, you can count by rational numbers if you want. I don't think counting is a good intuition.
On the contrary - to be countabley infinite is generally assumed to mean there exists a 1-1 correspondence with N. Though, I freely admit that another set could be used if you assumed it more primitive.
Isn't this what I just said? If I am not mistaken, this is exactly what "unique up-to bijection" means.
Anyways, I mean either starting from 1 or 0, they can be used to count in the exactly same way.
I'm arguing from the standpoint that we establish the idea of counting using the naturals - it's countable if it maps to the naturals, thus the link. Apologies for the lack of clarity.