[alain94040]

But this is begging the question. The author uses the definition that numbers have infinite decimals, and all you are allowed to do is ask for decimal n to find out a number. You can never grasp the full number, since you are given a finite amount of time (compute) to discover an infinite sequence of decimals.

From that definition, it's quite obvious that everything you compute has to be continuous, because you are never sure of what other decimals may be coming up, so whatever you compute has to be close enough.

That sounds more like an argument that representing numbers that way is not particularly useful since you can't do much with them (you can't even provide an equality operator).

[edflsafoiewq]

A constructive response is that the ability to examine a real number to arbitrary precision is already highly idealized. In the real world you will quickly exhaust your ability to measure a real quantity to ever higher precision.

> you can't even provide an equality operator

If you are given two rods, there is no way to tell if the two rods are of the same length.

[throwawaymath]

I'm not sure I'm following you. I'm also not arguing for or against the results here. I'm just giving you the background to understand the author's point; it's not something they just came up with, it's been under study for quite a while in constructivist mathematics.

I also don't really think you're using the right definition of computable here. You make it sound as though we're estimating, or truncating uncomputable numbers to make them computable when you say:

> From that definition, it's quite obvious that everything you compute has to be continuous, because you are never sure of what other decimals may be coming up, so whatever you compute has to be close enough.

It's not about being close enough or estimating, they're categorically different things. You can't obtain an uncomputable number, even by estimating, to any meaningful precision with a finite amount of time. So what are you saying here?