Neither the reals nor the naturals “exist”. We produce them as we need them.
Reals and naturals both have the same number of symbols: unknowable (“infinity”).
> Feel free to give me a generator function and I will give you a finite number that it cannot reach in finite steps.
f(n)=n
If you give n to that formula, it will generate n in 1 step.
> On the other hand, try giving me a finite positive integer that I cannot reach in finite steps.
Can’t come up with one. As long as is finite you can always just output the number itself.
Anything finite is countable. Anything infinite is unknowable.
Nono. F(n) where n is the number of steps, not the desired number. You’ve got it backwards. The number is the output, not the input.
That is why ascending order is the norm for the positive integers. For any positive integer X, it is clear that we will reach it in X steps using that generator function. So, it’s not a matter of being mainstream, it’s just the simplest.
The difference between the infinity of the reals and the positive integers is that there is no such generator function for the reals. No matter what function you give me, I can write a finite number that you will not reach in a finite number of steps.
Feel free to give me a generator function and I will give you a finite number that it cannot reach in finite steps. On the other hand, try giving me a finite positive integer that I cannot reach in finite steps.