If spacetime is continuous you effectively get infinite precision -> information density - at every point (of which there are an infinite number).
This seems unlikely for a number of reasons.
This doesn't mean spacetime is a nice even grid, but it does suggest it comes in discrete lumps of something, even if that something is actually some kind of substrate that holds the information which defines relationships between lumps.
> If spacetime is continuous you effectively get infinite precision -> information density - at every point (of which there are an infinite number).
This would be true if objects existed at perfectly local points. However we know that a perfectly localised wavefunction has spatial frequency components that add up to infinite energy. Any wavefunction with finite energy is band-limited. At non-zero temperature the Shannon-Hartley theorem will give a finite bit rate density over frequencies, and since the wavefunction is band limited it will therefore only have the ability to carry a finite amount of information.
This and the comment under it are making my point for me. Relativity assumes spacetime is continuous. Quantum theory implies that quantum phenomena are bandlimited and therefore the information spacetime can hold is limited.
The difference is we know what the quantised components of field theory are. We don't have any idea what the quantised components of spacetime are supposed to be, or how they operate.
The various causal propagation theories (like causal dynamical triangulation) may be the first attempts at this, but it's going to be hard to get further without experiments that can probe that level - which is very difficult given the energies involved. Without that, we're just guessing.
I often wonder if we simply constructed math the wrong way around.
People tend to mentally construct the natural numbers from set theory, wholes from naturals, rationals from wholes, reals from rationals and so on.
But what if there is some universe (in the math sense), which is actually complete and decidable, it's just that the moment you take discrete subsets of it, you also remove the connections that make it consistent or complete.
The very act of formalising mathematical concepts into words and paper is a quantisation step after all, because both are symbols.
Maybe there are proofs that can be inuitioned about (assuming brains are continuuous in some sense) but neither verbalised nor formalised.
> But what if there is some universe (in the math sense), which is actually complete and decidable, it's just that the moment you take discrete subsets of it, you also remove the connections that make it consistent or complete.
Presumably this hypothetical universe implements arithmetic, so it’s not complete.
Incompleteness means that there are true statments for which there are no proofs.
But that doesn't preclude proofs that are beyond the proof system that you proofed incompleteness for.
A non discete/symbolic proof might exist after all.
It would therefore not be the existence of the natural number subset that causes undecidability, but the missing parts of the non-natural superset required to talk about the proofs that cause undecidability.
The incompleteness proof comes to a russel paradox like contradiction, caused by a sentence of the form "I am not proovable." encoded via goedel numbering onto peano atrithmetic.
But proof by contradiction itself is problematic, because it relies on the law of the excluded middle, which only holds in classical two valued logic.
If you construct math from the top down rather than from the bottom up, then it is natural that it also has a multi valued logic.
In fact, it also would have infinitely valued logic. Infinite sentences, infinite theorems.
Such math is non expressible for us, because we rely on discrete descriptions.
But that doesn't preclude its existence.
Why should it though? What evidence do we have, if we can't express it or fit it into our existing mathematical framework.
To illustrate, think of Goedelsz approach and turn it backwards for a second.
Instead of taking predicate logic and assigning each sentence a natural number, imagine that predicate/classical logic is a different view on the natural numbers.
Now that means that there might also be logical interpretation of the reals, the hyperreals and so on. (We could do the same with Alephs, in fact they might be the more fundamental objects.)
Indeed! I suspect/realize that the incompleteness proofs are in fact artifacts of our discrete symbolization - a discrete symbolization that though has been incredibly effective about reasoning about continuous 'phenomena'.
Perhaps that is exactly what mathematics is? and all it can be? Or perhaps there is a 'higher mathematics' that we cannot reach yet? (or ever?)
Yeah, it's frustratingly difficult to investigate these avenues of thought, because they are almost by definition out of our grasp. Even worse the barrier to esoteric pseudo-science is very thin.
How do you know arithmetic is not complete or consistent? In ZFC arithmetic is complete and arithmetic is consistent. This assumes that ZFC is consistent. The second order Peano axioms are categorical so I assume you mean only the first order theory.
At any rate, what does any of this have to do with information capacity in the universe? Is the information capacity of the universe related to the consistency/completeness of arithmetic?
> How do you know arithmetic is not complete or consistent? In ZFC arithmetic is complete and arithmetic is consistent. This assumes that ZFC is consistent.
Aren’t you just begging the question by assuming ZFC is consistent to demonstrate that arithmetic is consistent?
I don’t think so. The point is, saying “arithmetic is inconsistent” doesn’t mean anything without talking about where this theory resides. The larger point is that this has no relationship to whether or not the universe is infinite so it shouldn’t be talked about at all within that context.
Suppose for a moment that it makes sense to say arithmetic is part of the fabric of the universe (whatever that is supposed to mean). How would one know if arithmetic is consistent or complete within the context of being part of the universe? Suppose ZFC is part of the universe. Then arithmetic (the model of it as being part of the universe) is complete and consistent. Now what? I claim nothing of consequence follows from this in relation to whether or not space is continuous.
Arithmetic is complete in ZFC. Well, in each model of ZFC sits a model of arithmetic (first order Peano axioms) and that model is complete. The incompleteness theorem doesn’t apply in this case because the incompleteness theorem is a statement about the first order Peano axioms and not about the situation in which they are residing in a larger theory (which in our case is ZFC). The Peano axioms are not able to prove their own completeness but if they reside in a larger theory then that larger theory may be able to prove their completeness.
If ZFC (or some other theory) implements arithmetic, then the first incompleteness theorem says that if ZFC is consistent then there must be true sentences in ZFC (not necessarily sentences of arithmetic) that can’t be proved in ZFC. Correct?
Yes! ZFC can’t prove it’s own consistency or completeness but a larger theory can do this. I think ZFC + Inaccessible Cardinal can prove ZFC is consistent.
I had two points in these posts. One is that none of this pertains to whether or not space is continuous. The other is that the statement “arithmetic is consistent” is provable in some contexts. It depends on what the actual theory one is dealing with. In PA it’s not provable but in ZFC it is.
If the universe “contains” a model of PA then is that model consistent or not? How does one know? (I doubt it’s meaningful to say that the universe contains PA though.)
> I had two points in these posts. One is that none of this pertains to whether or not space is continuous.
To me, “space is continuous” seems like a proposition that must be demonstrated a posteriori and I wouldn’t expect properties of formal systems to serve as evidence for the claim. So I agree.
> The other is that the statement “arithmetic is consistent” is provable in some contexts.
Agreed.
> If the universe “contains” a model of PA then is that model consistent or not? How does one know? (I doubt it’s meaningful to say that the universe contains PA though.)
Okay, yes, how can we interact with or measure PA as implemented by the universe? How do we (or can we) meaningfully talk about the universe implementing PA?
Yes those questions seem interesting to me, but I don’t have anything intelligible to say about them.
> If spacetime is continuous you effectively get infinite precision -> information density - at every point (of which there are an infinite number).
Even if space is continuous, that doesn't mean we can get information in and out of it in infinite precision.
Look at quantum physics. Maxwell's equations don't suggest existence of photons (quantized information). But atoms being atoms, they can only emit and absorb in quanta.
Obviously that "something" is the float type used to calculate the simulation. Probably some ultra dimensional IEEE style spec that some CPU vendor intern booked anyways. ;)
Im pretty sure my math is right here, so… It would be more appropriate to say they failed to shell out for the 256 bit processor. Because at 256 bits per int a vector of 3 ints can easily encodes any location in the observable universe as Planck length coordinates.
Ya, but the observable universe only exists for cache locality reasons. You dont want to have to transfer information across too many processors as your bus bandwidth limited. The actual universe is much larger.
We should be careful then not to overload the 256 range. If we probe too closely at the QM level or use too many quantum computers it might overload the local processor node and crash it. Be a bad day for everyone.
This seems unlikely for a number of reasons.
This doesn't mean spacetime is a nice even grid, but it does suggest it comes in discrete lumps of something, even if that something is actually some kind of substrate that holds the information which defines relationships between lumps.