Which would mean integers are baked in, rationals too, but non-constructible reals (essentially all reals, given any degree of approximation) are a useful abstraction but don't actually exist in any way.
Reals are not real.
(Roughly) Equivalently: There may be no perfect circles in nature.
Constructible reals are also a subclass of reals, but you can't claim anything about the class of reals, which are vastly dominated by un-constructible reals, because constructible reals have a property.
There are many reasons to doubt un-constructible numbers exist in nature.
Just for starters, you can never actually define a specific un-constructible real. If you did, you would have defined it, making it constructible.
An un-constructible real requires infinite information to define. Not infinite digits (pi is constructible, e is constructible), but an infinite list of uncompressible digits, or some other expression with infinite numbers of symbols!
The name "reals" is highly deceptive/unfortunate. (What could be more reasonable than a "real" number?)
We need a pithier name for constructible numbers, and that is what should be introduced along with algebra, calculus, trig, diff eq, etc.
None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration.
(It would be nice to rename "real" numbers, to mean actually real numbers that we could actually use. But given the generations of confusion that would incur, I propose "actual numbers", to be all constructible numbers. Nobody but mathematicians, who play abstract games with higher order infinities, need "real" numbers.)
Well no, you will never have it.
You can't start out with finite things, and built an infinite thing, even if you have infinite components to put together, and infinite time to do it.
That is what countably infinite means. It is a very practical kind of infinity.
And the concept comes directly, and inevitably, from the integers.
Just like integers, and all the theorems/patterns we discover in them, reality may be countably infinite. Filled with an infinite number of structures of unbounded sizes, and infinitely large structures parameterized with finite constraints.
The class of real numbers is not what our familiarity with its name makes us think it is. It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat. Not by induction or other mathematical inevitability. Nowhere in the chain of numbers built up from the integers. With no hope of ever encountering a single concrete instance that isn't already in a smaller better defined subclass, that doesn't require the concept of reals.
Mathematicians get to have their games. At a minimum they are useful as ways of stretching mathematical skill. Concepts that don't correlate with things that exist, can still be interesting, challenging, and spin off insights.
If you accept countably infinite rationals then you also accept Cauchy Sequences, no? Then we see that the reals arise naturally from rationals. I'm a noob at this btw so I would appreciate guidance.
Absolutely.
When most people think of reals, they are thinking constructible reals. Which are countably infinite in number.
If there is an equation (known or in principle), they are constructible.
If there is an algorithm (known or in principle), they are constructible.
Limits, continuous fractions, all those are constructible.
From a mathematics perspective, I think it's a loss that the distinction between countable/constructible numbers vs. uncountable/un-constructible is completely blurred under one name "reals" early in everyone's math education. Even though the difference is significant when reasoning about information, the relationship between math and physics, math and computation, etc.
And about infinites. Most of us famously know that there are infinitely more reals than integers.
But how many people know that all well defined reals, constructible reals, calculable reals, and their equations, probably everything they imagine when they think of "real" numbers in practice, remain countably infinite. Exactly like the integers.
The set of constructible reals is the same size as the set of integers.
Brownian motion?
Similar to how we model pressure, temperature and volume relationships of gases, irrespective of the individual particles they represent.
But in principle, with enough computing power, quantum field equations can model the same phenomena at the particle level.