the natural numbers (1,2,3,4 . . .) are countably infinite - this "size" infinity (cardinality) is called Aleph_{0}

the real numbers 0, .00000000001, pi, 23, 1234.56789, 123456789, and a bajillion-bajillions, literally every number (and every number in between) on the continuous number line are uncountably infinite - it's cardinality is Aleph_{1}

I think the idea of ~0 being a "smallest possible non-zero number" tries to reconcile the paradoxical uncountably infinite places to stop on your way from 0 to 1. sort of like a Planck's length in physics.

However, I would argue that there would be a one-to-one mapping from the number line constructed from multiples of ~0 and the natural numbers. so this number line would also be countably infinite and really, no different or any more useful than the natural numbers already are.

there could be a possibility of some sort of fractal-countability here (sort of like how certain objects are fractally dimensional - more than 1, less than 2) where a number line might be constructable that has cardinality between Aleph_{0} and Aleph_{1}.

If there is, that's above my pay grade, but 0 and infinity have this sort of black hole type property when it comes to countability and things kinda get pulled one way or another so I don't think this is really a possibility.