> So now you've got a binary fraction that most would say specifies a real number. An almost surely non-constructible one in fact.

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.

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threatofrain 4 days ago | parent [-]

> It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat.

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.

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Nevermark 4 days ago | parent [-]

> If you accept countably infinite rationals then you also accept Cauchy Sequences, no?

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.

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throwaway81523 2 days ago | parent [-]

> If there is an equation (known or in principle), they are constructible.

Brownian motion?

	▲	Nevermark 2 days ago \| parent [-]
		Models of brownian motion are highly accurate approximations that describe how many particles behave together in summary form. 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.