The difference between infinities is I can write every possible fraction on a piece of A4 paper, if the font gets smaller and smaller. I can say where to zoom in for any fraction.
I can't do that for real numbers.
I can't do that for real numbers.
The more I learn about this stuff, the more I come to understand how the quantitative difference between cardinalities is a red herring (e.g. CH independent from ZFC). It's the qualitative difference between these two sets that matter. The real numbers are richer, denser, smoother, etc. than the natural numbers, and those are the qualities we care about.
I don't think this definition is that weird, for example by 'larger' I might say I can easily 'fit' all the rational numbers in the real numbers, but cannot fit the real numbers in the rational numbers.
So, let's inspect pi. It's a fraction, precision of which depends on how much you zoom in on it. You can take it as a constant just for having a name for it.
The zooming is finite for every fraction.
How?
> Problem is with the assumption that some numbers such as pi or sqrt(2) can have absolute and full precision.
That's the least of your problems, I couldn't even draw different sized splodges per real number.