4 ms·
Wait, do the natural numbers and rational numbers have the same ordinality? It seems not because of the whole 'a number in between every 2 rational numbers'. Bu
by rocqua 9y ago
Wait, do the natural numbers and rational numbers have the same ordinality? It seems not because of the whole 'a number in between every 2 rational numbers'. But then what is the ordinality of the rational numbers?
- Veedrac 9y agoThey have the same cardinality, but there is no order-preserving bijection.
- bmm6o 9y agoThe standard argument for showing that they have the same cardinality is to consider the matrix (let's see if I can render this): 1/1 2/1 3/1 4/1 ... 1/2 2/2 3/2 4/2 ... 1/3 2/3 3/3 4/3 ... ... You can traverse each SW-NE diagonal, producing the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, ... This will eventually hit every positive rational (several representations, in fact). If you want, you can start with 0 and insert -q when you hit q.