Godel's first theorem says no formal system with recursively enumerable axioms and which is powerful enough to express elementary arithmetic can be both consistent and complete. Most people adopt the response – alas, that means no formal systems we can devise can ever be complete (except for systems too weak to express arithmetic, which are not very useful). But, paraconsistency and dialetheism give us another response: yes, we can have complete formal systems with recursively enumerable axioms and which are powerful enough to express elementary arithmetic. Okay, they'll be inconsistent, but inconsistency is not as bad as you thought it was. We can contain the inconsistency to some small area in which it isn't going to bother you, you'll scarcely notice that it is there. We can hide the contradictions in the closet and go on our merry way not thinking about their existence except on rare occasions.

If mathematics is inconsistent, does that make mathematics objectively unreal? Only if we insist objective reality must be consistent. If objective reality contains dialetheias, then inconsistent mathematics can be objectively real.

EDIT: see also https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...

I'm assuming we're talking about Gödel's incompleteness theorems?

Don't they just say that if there's such a thing as objective mathematical reality, it can't be effectively axiomatized?

Well, math is a universal approximator -- if there are patterns in what we observe of reality, math can fit them, but that's a far cry from them "being" math. Other formal systems like Turing machines or Lambda calculus can also approximate anything (including each other, naturally) with different primitives... and they have an easier ontology (as far as I can tell). I mean, if you posit that fundamental reality is a computer of sorts, you get your ontology, and you get math as a very good formalism to describe the particular program we're in.

No they don't. Real space doesn't appear to be infinite, and Zn is not subject to Godel's incompleteness theorem.

If you drop the requirement of infinite numbers and "recursive" infinites (e.g. real numbers), as reality appears to do, there is no problem.

Citation needed. People keep saying things like this ("real space isn't infinite" or "real spacetime must be discrete") but never show their work.

That being said, nature has given us very strong hints at discretization. For example: https://en.m.wikipedia.org/wiki/Planck_length. Because of uncertainty principle, we can't measure distances less than that. In a similar way, time also appears to be discretized, along with energy and mass.

From the article:

> Importing the atomic philosophy of Democritus into modern physics might be essential for reconciling general relativity (which assumes a continuous reality) with quantum mechanics (which very much does not).

As far as infinite vs finite space: because there is a fixed rate of information propagation, we only need to show that time is finite in extent, and it follows that space is also finite. We know that the visible universe is finite (by looking in the sky) and we can explain why: the big bang happened at a finite time in the past.

We don't know if the universe will have a finite future, but it might be the case. One possible cause of a finite future could be accelerating expansion of the universe due to dark energy. [2]

Of course there are other interpretations of "real space", but this one (causally connected) seems parsimonious to me.

[1] https://news.ycombinator.com/item?id=13466254

[2] https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe

That does not follow: As our universe is spatially flat (or close to it), it is consistent with the infinite-space Friedmann model. The cosmic horizon that bounds the observable universe would still exist due to the metric expansion of space.

Just to clarify, FLRW is a class of models, which includes ones with finite as well as infinite space. We've been able to determine that our universe is close to spatially flat. However, that does not mean it's infinite: If spatial curvature is positive but small compared to the cosmic horizon (which is due to spatial expansion), we won't be able to tell.

That said, even given finite space, spacetime can of course still be infinite if the universe does not recollapse.

Your criticism is right and that is a bad argument. The real reasons they are hard to reconcile are technical (see for example [1]), and not mentioned in the source I referred to.

The article only says that a discrete theory of spacetime "might be essential" for the reconciliation; the arguments for this are philosophical - it would be beautiful and historically completing. Read the article if you're interested in the arguments, which aren't included in my comments.

> And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model, and is supported with high accuracy by the latest experimental observations.

I agree with you that the universe could be infinite, but that doesn't contradict my previous comment. I explained that I was talking about local (causally connected to us) space-time, only. I admit this is a bit of a trick, but the parent comment used the term "real space" which I interpreted in such a way to favour an argument for finite space.

> So again, your argument doesn't seem that compelling to me.

I've given reasons but I think it might be helpful to restate what I am actually claiming. I admit I don't know that space and time are discrete. The source I gave is a poetic and philosophical argument that it might be. I think it's established fact that energy and matter are, and that time is finite in the past, so the visible universe is finite. I think it's possible that the future is finite too.

I don't know what FLRW is. My best guess is https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–W...

[1] http://www.theory.caltech.edu/people/jhs/strings/str115.html

None of that is established, actually:

Metric expansion of space comes with a cosmic event horizon where redshift goes to infinity. While the energy contained within that region is finite, there's no reason to assume that the universe outside the horizon looks any different than inside the horizon, and it might very well be spatially infinite.

As to time being finite in the past, it is possible to generalize Einstein's equations so that they can be extended past the big bang singularity, and what you get is a mirror universe on the other side. The issue here is that as we approach Planck scale, quantum gravity becomes relevant, and we have no predictive theory of that.

It's also worth pointing out that the original statement is nonsensical because the theory of real closed fields doesn't suffer from incompleteness in the first place, so there's no need to "save" it from paradox the way you have to save Peano arithmetic.

I do not see how that's true for quantum mechanics (take linear superposition as just one example). People may speculate about discrete underpinnings, but at this time, it's merely speculation.

Eg for the qubit, the state space is the Bloch sphere, which is continuous.

Furthermore, quantum computing and classical computing are known to have the same computational power, just different computational complexity, ie. any quantum system can be simulated by a classical system with exponential slowdown, at worst.

There are plenty of papers exploring this territory [1,2,3] if you want further details. There are older references but they're not easily available online.

[1] Computability in Quantum Mechanics, 1995, https://philpapers.org/rec/MYRCIQ

[2] Effectively calculable quantum mechanics, 2015, https://arxiv.org/pdf/1508.03879.pdf

[3] Constructive physics, https://arxiv.org/pdf/0805.2859.pdf

But this is exactly my point. The fact that our observations have finite precision doesn't say anything about the underlying reality. That's the part that you're asserting. If it also turns out that assuming the underlying system is continuous results in much more compact theories than assuming otherwise, why is the obvious conclusion that spacetime is in fact discrete?

I'm not sure what you think I've asserted, but my only actual assertions are that all of modern physics requires only computable abstractions, and thus, continuous abstractions aren't strictly necessary.

I can certainly make the case that a continuous ontology requires more ontological commitments than a discrete ontology, but I haven't said anything along those lines thus far.

> If it also turns out that assuming the underlying system is continuous results in much more compact theories than assuming otherwise, why is the obvious conclusion that spacetime is in fact discrete?

What is the meaning of "compactness"? A proper comparison of theoretical parsimony requires comparing the Kolmogorov complexity. Compressing ontologies positing reals and/or rationals are definitely less parsimonious than ontologies positing only naturals.

As far as I can tell, by this argument there is literally nothing that would convince you that anything but a discrete theory was correct. So I don't think it's a very interesting argument, sorry. But thank you for explaining your viewpoint.

My point is that at worst, the incompleteness theorems only imply that we won't be able to write down all the rules that govern the universe.

Not being able to write all the rules is a completely different and unrelated thing.

So, assuming the initial system is consistent, adding the godel sentence, or its negation, produces a new consistent system, iirc. But in the second case, it would be omega-inconsistent ?

Assuming the original system was consistent (which you can never prove), yes.

But that doesn't help you, because the new system with these additional axioms will still, provably, contain statements that can't be proven true or false within that system (or else be inconsistent). (Turing did try some programme where you keep adding new axioms and take the transitive closure over all the axioms you'd add, but AIUI that doesn't work out either).

Careful there: Gödels first incompleteness theorem proves that in a consistent effectively axiomatized formal system, there will be statements that are true or false that cannot be proven to be true or false.

So let's assume there were such a thing as objective mathematical reality and that it could be formalized as The System. As far as I'm aware, all the incompleteness theorem implies is that The System's set of theorems is not recursively enumerable (ie we will never be able to 'write them down').

All we can do is craft necessarily incomplete subsystems applicable to areas of our interest. It's a bit sad that humans will never know the Whole Truth, but such is life...

While you no longer have undecidability (and therefore no incompleteness, either), you do have its finite form, namely infeasibility (due to computational complexity), which, given an assumption that time is finite, amounts to precisely the same result. Almost every theorem about undecidability can be generalized to infeasibility. For example, instead of undecidability of the halting problem, we get infeasibility of the bounded halting problem (that's the basis for the hierarchy theorems, which gave birth to the notion of computational complexity, just as the halting problem gave birth to the notion of computability). Similarly, by bounded halting, instead of a consequence of incompleteness, i.e. "there are statements that can neither be proven nor falsified", you get "there are statements that can be neither feasibly proven nor falsified" -- this is a direct corollary of the hierarchy theorems -- and the problem remains the same. The only question is whether you're willing to admit infinity (or what kinds of infinity you're willing to admit) as a convenience, or as an approximation of large finite quantities. But choosing not to admit infinity doesn't really make anything simpler (on the contrary, things may get more tedious).

Logically, we are not finding better and better approximations to some mathematical laws of nature, but rather making a series of arbitrary choices (you can assert the truth or falsity of any independent statement). Philosophically, we can argue about the sense in which the continuum hypothesis ought to be true or false (that is what Gödel did), but practically it makes more sense to think of logic as a tool that you build to describe and solve specific problems.

[1] https://en.wikipedia.org/wiki/Hilbert's_program#Hilbert.27s_...

Presumably this is an ever-present hazard in any proof using the axiom of choice - in this case it is relatively obvious, but how easy is it to accidentally rely on such an uncomputable object in a dusty corner of your proof?