But in our universe, if it's discrete on a macroscopic level, then it's either inherently a natural number (it can be counted), or it's intelligently created.
I'm a little uncomfortable with the language that the theories "say that the world is" X. General Relativity and the Standard Model both model the world using real numbers, but they're both known to be wrong, and the fact that they are continuous is not a great reason to claim that the universe is continuous.
On the other hand, observations about Lorentz symmetry holding at distances on the order of the Planck scale put a wrench in a bunch of discrete spacetime theories. I don't really understand the math, though.
All this is somewhat tangential to the issue of real numbers. Real numbers are not necessary for continuity.
I think it is easy to get around the problems of Lorentz invariance in discrete models, as long as it is not the spacetime that is explicitly discretized on the lattice. The lattice must be some other combinatorial graph structured algebra, with non-local 'propagation' of fields. Quantum Mechanics says that space is only defined relationally on the intervals between field interactions: a 'particle' is only localized as a particle when it interacts (position or momentum observable). So discrete space and time appear as values on some subset of lattice nodes in response to some propagating fields ('particles' only at the interaction event). The slogan for this is 'spooky distance at an action', because it is the (inter)action that defines the space(time).
QM (and QFT) assume a background time, it is not an observable, even though it appears to commute with energy (e.g. energy is momentum in the time direction). So it's more tricky to understand how time emerges in a discrete Quantum Gravity, but I suspect there is an intrinsic proper time, defined by interactions with the 'vacuum' (minimal field states on the underlying lattice), which bootstraps a relational time defined over intervals between interactions.
Let's avoid equivocating here: "the continuum" is sometimes used to refer to the real numbers, but "continuity" in this context is a property of functions between metric spaces (or possibly topological spaces). "The continuum hypothesis" and "continuous functions" are actually from completely different branches of mathematics.
This happens fairly often in mathematics, where similar-sounding terms are used to describe completely different concepts, or the same term sometimes means different things in context, or sometimes an Adjective Noun is neither described by Adjective nor by Noun.
We have the holographic principle from physics which says that the amount of information in a finite volume of space is discrete.
If true, then we can think of the continuity just as a convenient interface. All experimental questions can be answered by a simulation with a discrete state space.
General relativity and the standard model also produce infinite values, which to me is a pretty sure sign your model is broken. Viewing black holes as infinitely dense hasn't yet caused us problems in terms of predictions (but they're still a bit wild west area), and they fixed the standard model with renormalization (which seems like a bit of a hack). They're good enough to be useful but I'm pretty sure everyone expects them to be radically reformulated at some point.
Do they really say this? They assume this just as Newtonian mechanics does. And like Newtonian mechanics give reasonable answers in the domains they describe. How would they change if there was a cutoff at some infinitesimal scale?
It is definitely far, far more than just what particles exist. The Standard Model describes how those particles interact with via the three forces other than gravity. The particles themselves are modeled using quantum fields and the properties of particles arise from operators on those fields. Those operators don't work unless the field is continuous.
The operators will have a spectrum which describes how the corresponding observable is quantized.
That's not really ELI5, but remember learning derivatives in calculus? Just like you can't take the derivative of a function which isn't continuous, you can't use the Standard Model if spacetime isn't continuous.
I simply can't make heads nor tails of the parent comment. It was probably downvoted because people think it is nonsensical. Here is my explanation:
1. What does it mean to be "discrete on a macroscopic level"? (If a universe were discrete, would it not be discrete at any scale? When we use the term "discrete", are we talking about state space, or discrete spacetime, or what?)
2. What does it mean for a universe to "inherently [be] a natural number"? (Universes are not numbers, right? Is some sort of claim that the universe's state space is finite?)
3. What does have to do with whether a universe was "intelligently created"? (How can it possibly make sense that "intelligent creation" is an alternative to a discrete universe? The claims seem entirely unrelated.)
The "explanation" comment doesn't attempt to explain anything so I'm not sure why you call it an explanation. (Well, perhaps it explains that a conditional with a false antecedent is logically true, but most people here already know that, so pointing it out is not a great way to contribute to the discussion.)