I know what the paradox is, but how does it relate to our universe?
In: Mathematics
> Are space and time infinite?
ELI5 Answer: We don’t know, but probably not. More on that in a second.
Our mathematical models work in an infinite, real valued spacetime, but that’s different than us knowing what spacetime actually fundamentally *is*, what the true physical structure of reality is. Physicists are divided if either space or time are fundamental physical features of reality, or just emergent properties, epiphenomena of some deeper underlying structure. As the saying goes, “Don’t confuse a model of a thing for the thing itself.” The model is just a good story, an attempt to impose an elegant mathematical structure onto the physical phenomena we empircally observe.
For all we know, space and time could be discrete (unlikely, but possible) rather than real-valued, and / or finite. Just because you can define an infinite coordinate system doesn’t mean the physical universe extends infinitely in every direction. The laws of GR work just as well anyway in a flat or hyperbolic spacetime extending infinitely as in a (hyper)spherical spacetime that “loops” back on itself and occupies a finite (hyper)volume.
Besides the fact there’s no way to observe a physical infinity (there will always be a cosmic horizon which limits our view of the universe, and by definition you can’t verify something is infinite by observing a finite part of it anyway), there are philosophical grounds on which people might think the universe can’t be literally infinite, and one of them is the Hilbert’s Hotel thought experiment.
> Hilbert’s Hotel Paradox–how does it relate to the real world?
See, Hilbert’s infinite hotel is a thought experiment that highlights the very unintuitive properties of infinity, which (in most mathematical usages of the term “infinity”) is based on the idea of cardinality (there are other “measures” that are not cardinality, but that’s not ELI5), in which two sets have the same cardinality (have the same size, contain same amount of stuff) if they can be put in one-to-one correspondence with each other (in math terms this is called a bijection). This leads to various unintuitive results like the fact that the odd numbers, the even numbers, the prime numbers, etc. all have the same cardinality as all the natural numbers, and even all the integers, and even all the rationals!
And indeed, it leads to results like if you had an infinite hotel room, it could be fully full with no vacancies, but you could still make room for a countably infinite number of new guests, even an unbounded (finite) number of busses each carrying a countably infinite number of new guests.
Now that fact is where philosophically we start to suspect the mathematical concept of infinity might not correspond to the real, physical universe. Because while it makes perfect sense mathematically, it seems physically absurd.
Here’s the thing, infinity exists in math, in all modern set theories (e.g., ZFC) in so far as we choose a system with the axiom of infinity, and we can talk about sets in such a system and do math without contradicting ourselves (so far). But that’s pure abstract math—set theory shows you can talk about infinity without contradicting yourself. But it doesn’t not have to correspond to real life.
In fact, once you get into uncountable cardinals or in some axiom systems inaccessible cardinals, it seems completely divorced from reality—what would it even look like to have an uncountable amount of stuff in real life, or even an amount of stuff whose cardinality was an inaccessible cardinal?
Hilbert’s Hotel shows us that if an actual literal infinity of things existed in real life, it would have very bizzarre implications. Which suggests while we can talk about infinity in our axiomatic set theories, it probably doesn’t exist in real life unless you want to admit various physical absurdities.
Hilbert’s hotel isn’t really a statement about our universe per say. It’s just a made up word problem to illustrate a property about infinity. Kinda like teaching division by using a made up problem about sharing a plate of cookies equally among a group of people.
Would you say that the cookie example tells us something about the universe? Not really, they just help you visualize a circumstance in which you might care about division.
In a similar way, Hilbert’s hotel is a circumstance where we might care about adding two infinite things together. It’s just that infinity behaves in weird ways, so the solution to Hilbert’s hotel seems like a made up solution to a made up problem. Infinity plus itself equals… itself. Exactly the same as itself. “Real” objects don’t typically work that way, so we have to make up this hypothetical story to make it digestible for laymen.
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