Yes, apologies -- using the term 'construct' muddied the point as sqrt(2) is a 'constructible' number as you point out. The term 'real' is what's at issue here and I am arguing for a distinction between a 'map-like' real and a 'territory-like' real, the latter of which has some sort of spatiotemporal grounding.
> There are some numbers that are not, and perhaps these can truly be said to not exist.
So then we have a real issue because the vast majority of the real line is composed of these uncomputable numbers which you've suggested don't exist.
Sure, but no one uses the reals anyway for 'constructible' things. As I pointed out, the computable reals themselves form a closed field and are the things you would find when describing real life.
As for the 'problem'... I personally don't view it that way. In my opinion (and it's just that, since there's no mathematical 'truth' here), I don't believe non-computable reals exist in any meaningful way. I believe this is similar to how we talk about a 'program that can check if another one halts'. Anyone can make that statement and claim that such a thing exists, but it's not at all clear that such a thing exists. But that's a lot different than saying there are X particles in the universe, thus the number X + 1 does not exist. Because x + 1 does exist and you can write a turing machine that can compute it to any precision (or a lambda calculus function that'll give you the next church encoded representation of it, etc).
My point is two fold. Firstly that there are certainly numbers that are greater than the total number of 'stuff' in the universe. Secondly, that there are some numbers that cannot be described in any meaningful way. These can be said to not exist (my belief), but others disagree.
> There are some numbers that are not, and perhaps these can truly be said to not exist.
So then we have a real issue because the vast majority of the real line is composed of these uncomputable numbers which you've suggested don't exist.