/sci/ - Science & Math

What's the error in me thinking:

[math]A[/math] countable, so can write it as [math][a_0, ..., a_n, ...][/math].
[math]<_A[/math] an order, so wlog [math]a_0 <_A a_1 <_A ... <_A a_n <_A ... [/math].
Then [math]a_i \mapsto i [/math] is the required map and [math]S = \mathbb{N}[/math].

It seems a bit too good to be true