Our universe is most likely a computer simulation

[daaron]

145. I get what your saying, I think.

Again, the answer seems to be another round of "Yes," with the usual caveats. I don't know much about quantum loop gravity, but suspect it attempts to reconcile sub-Planck wavelength model with GR by force (of gravity), rather than the spacey stringy approach - but any contender for a ToE has to reconcile QM & GR, by def. Aristotle commented on the math problem involved, which play footsy with the paradoxical - infinitely divisible (Zeno's 'paradox') VS infinitely extensible (ie, counting to infinity). The real number line combines these two infinities: it includes all the countable numbers up to aleph-0, plus all the rational numbers (fractions) up to aleph-1, plus all the irrational numbers that are infinitely dense between any two points on the line.

Then there's the outliers - the square root of -1. What do we do with that? It turns out the reals are algebraically extensible. They happen to extend in fact to the complex number field, where dwell dragons and other imaginary numbers. In fact, we can represent the complex numbers by an extension field of two dimensions. Hence we can write a basis as {(0 1), (1 0} and perform various feats such as rotation and reflection in 2-D. It is by mere convention that we write the complexes as z = x + y*i. Here, the "i" acts as an unknown, not a variable, and any transcendent number, such as Pi, can fill that formal role as well in algebraic computations. So these additional dimensions are identical to lines - the axes if you will - each of which is a real number line.

There's continuity for ya. Roughly - pick a point, any point, in any vector space. Draw an 'open' sphere around the point on your curve (imagine a string) of radius epsilon. That's the epsilon-neighborhood (E-hood). If you can shrink epsilon down to arbitrarily close to zero radius, and there is always another point in the E-hood, you've got continuity. If suddenly you have an empty E-hood, your curve is not continuous. There's a missing point. General relativity needs this continuity, for the maths to work out. QM, OTOH, deals in the discrete and discontinuous, with a natural minimum distance of the Planck length being the diameter of the smallest particle (since a shorter wavelength creates a different particle, altogether, these don't exist, per se).