I remember when I was young and first learned about prime numbers. I thought, how many of them are there? I asked my teacher and he said they go on forever and pointed to the following famous proof in a book:

Suppose that p1=2 < p2 = 3 < … < pr is all of the primes. Let P = p1p2…pr+1 and let p be a prime dividing P; then p can not be any of p1, p2, …, pr, otherwise p would divide the difference P-p1p2…pr=1, which is impossible. So this prime p is still another prime, and p1, p2, …, pr would not be all of the primes.

Being young, this took some effort to understand the concept expressed by the language.

It depends exactly what you mean by a “statement about the natural numbers”, but if you are satisfied by the translation as “a statement of first-order arithmetic” (i.e., a statement that can be written with equality, the operations + and × (you can throw in power, it won’t change anything) and all quantifiers ranging over the set of natural numbers), then the answer is no: every statement of first-order arithmetic that can be proved using ZFC (+GCH if you will) can, in fact, be proved in ZF.

The way this (I mean, the meta theorem I just stated) is proved is by using Gödel’s constructible universe, which is a class of sets L, definable in ZF, which is a model of ZF, in which the axiom of choice (+GCH) automatically holds; and this class L has the same set ω of natural numbers as the universe. So if you can prove something from ZFC(+GCH), it is true in L, and if it is an arithmetical statement, then it speaks about ω which is the same in L as in the universe, so the statement is true in the universe.

Decoherence doesn’t require a human knowing about it. Spacetime represents our reality and converts virtual quantum information to physical/real objects.

Observation/Measurement is dead. Spacetime determines if a quantum wave should be physical for our reality. Spacetime governs our reality, by handing out physical states. Time dilation demonstrates spacetime scaling reality.

The flight/path/state of a particle/wave is known before starting. If a spacetime object (detector) causes the particle to decohere but continue moving to a final panel, the particle/wave is given a physical state from the start. If the particle/wave is to pass two detectors before the final panel, the particle/wave starts as a wave ..the physical state is taken from it.

Are unobserved matter waves, virtual mass in a 4D format – without time (don’t age/decay)? When it is given time it becomes physical in 3D and the 4D is used for time? The temporal dimension is where the fabric of spacetime originates, anything there is 4D by default. It isn’t spatial but mass can live there as quantum waves ..virtual.

A physical state turns a wave physical before it starts moving. It won’t be a wave during its flight.

4D virtual mass is unobservable. A physical state from spacetime is transforming the 4D to 3D + time.

Dark matter is unobservable, but also doesn’t have the ability to be given a physical state.

Does observation/spacetime swap quantum waves by giving it a physical state and a timeline? The wave function can propagate, but the wave doesn’t age until given a physical state.

Does this explain why we can never see quantum waves ..they are 4D?

There is no reason for giving “observed” vs “unobserved” particles any special properties. At most a measurement can make the wave function for the position more narrow which seems more particle-like. At the end of the day it’s always described by a wave function. Wave- particle duality is a relic from a time where we were initially trying to understand QM.

Dark matter is observable through gravity. Otherwise we wouldn’t have observed it. I know that’s a tautology, but it seems it needs to be pointed out.

Sure there is, unobserved particles can tunnel, entangle, and be in superposition.

Dark Matter is not directly observable, you aren’t going to see a particle of it.

“Unobserved” / “observed” is not a well-defined property of a wave function. You cannot tell whether something was observed or not observed. Observation simply changes the state of a system to an eigenstate of some observable, which is just another state.

Decoherence is the difference between observed and unobserved

There’s no way to make a measurement so a particle is literally at one point, hence they are always in superposition over positions. It’s just a matter of how spread out.

And even if we did accept there was such a difference, the particle would be in this “observed” state for literally a point in time and go back to being unobserved. It wouldn’t make sense.

Observed particles are not in superposition (they are not in a state that is considered quantum weirdness), they have uncertainty because the quantum field still has an influence on it.

The observed state lasts from point a to b. It’s given a timeline. If it hits an object too large to be influenced by the quantum field it remains observed as it is part of that object now.

I’ve often thought that the universe itself is incapable of storing a perfect memory of anything more than a tiny percentage of its contents–some laughably small fraction of its whole. Even with an amazingly efficient system, how much information could really be stored about what’s transpired in the past?

By this I mean: I’d like to know, for example, which oceans the molecule of water in my glass has been immersed in, which algae used it as part of their metabolic processes, which comet deposited it onto the proto earth, which nuclear furnace generated the oxygen that went into its formation, etc.

I’d like to know this for as much of the universe as possible, in as much detail as possible. How much of the universe would be required, and how much information could you store?

Here are my thoughts on how to solve this. Considering:

There are some 1e83 atoms in the universe. Describing an atom at a macro level would mean you’d need to store its position and velocity over time, integrated into arbitrarily small time units, or perhaps only recording the changes in velocity when it accelerates (but even then you’d be discarding true information regarding its precise movements from heat and possibly inscrutable stochastic processes if it could even be attained at that scale without destructively altering the atom as it goes about its normal business).

The atom itself isn’t even atomic, as there are quarks and other strange particles, but you get the idea of “atom” for the thought experiment.

You’d need some multiplicand of atoms to describe this information–very likely already requiring a few orders of magnitude of atoms per recorded atom.

This would be a reading and writing mechanism, again a few orders of magnitude per atom that somehow encodes information about the ones you’re observing. (I.e., computer memory requires some 1e25 atoms (500g of silicon) to manipulate 8e12 bits of information (1 TB)–criminally inefficient.)

You’d need motive forces that would be so inclined as to construct this system, themselves constituting overhead rather than a storage mechanism. E.g., powerful creatures with space thumbs.

You’d need to have constructed the system in such a way that it could transmit or collate that information despite the inflation and entropy of the universe, and be willing to discard or localize data (locally accessible but inaccessible from other parts of the universe) that simply couldn’t overcome time horizons.

You’d need a non-destructive or at least not particularly invasive method of recording the state of each atom even when squeezed in among siblings (and we’ll assume such a thing is possible, although it almost certainly isn’t)

In a Gödel-esque puzzle, as I vaguely understand the term from GEB, the system itself would necessarily fail to encode information about its own history

Thus in the best-case scenario, you’re talking about an encoding inefficiency that necessitates only being able to record the state of some smaller subsection of the rest of the universe, and that would require the entire rest of the universe being dedicated to that task, which is presumably unlikely.