[klodolph]

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.

[mikhailfranco]

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.

[ska]

All models are wrong. Some models are useful.

Some of them extremely useful :)

[maverick_iceman]

>Real numbers are not necessary for continuity.

You know of any continuum that doesn't include the real numbers? That will contradict the continuum hypothesis.

[klodolph]

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.