As I understand, that works too, and leads down that path to exploring the multiple 'levels' of infinity.

Why not ask them to move to room 2N+1? Then you could accommodate an infinite number of new guests (all the rooms 2N are now empty) before having to ask anyone to move again.

The important feature of Hilbert's example is not the hotel, or the guests, but the strange consequence of Cantor's method for 'counting' an infinite collection of objects. William Dunham's Journey through genius does an excellent job with Cantor in his final two chapters. I'm paraphrasing him.

Suppose you want to know if you have more than 5 raspberries. You could count them, or you could stick them on the ends of your fingers. If you have berries left over, there were more than five. If all the berries fit all the fingers, you have put your fingers in a 1-to-1 correspondence with the berries, and concluded that the numbers of each are the same. That's how small children count, and Cantor's genius was to extend this idea to sets of infinite objects.

If you can find a 1-to-1 correspondence between the counting numbers (1,2,3,...) and some other set of objects, then the size of the two sets must be equal (in the business, size of sets is called the 'cardinality'). It is traditional to denote the counting numbers by 'n'.. Hilberts 'paradox' arises from Cantors claim that there are as many counting numbers as even numbers. The correspondence is n->2n. For every counting number, I can double it to find every even number. That's weird.

You can just as easily show that there are as many odd numbers as counting numbers. The scheme is more complicated, but you can also find a 1-to-1 correspondence between the counting numbers and all the fractions between 0 and 1. It seems like the counting numbers can count everything! But that's not the whole story. The big shock is that there are more numbers between 0 and 1 than counting numbers. That is Cantors 'nondenumerability of the continuum'. I highly recommend Dunhams book. The proof of the nondeumerabilty theorem is so beautiful and accessible (like accessible to a ten year old).

I've seen the paradox mentioned a lot but I don't see it as paradox at all.

A hotel with infinite rooms has infinite size.

Just idea of asking guests to move would take infinite time (i.e will never happen).

if each guest wait for the his new room to vacate, no one would move at all.

No. Your answer has absolutely nothing to do with infinity nor is it paradoxical.

The paradox is saying that for every number N there is an occupied room in the infinite hotel. But that there is still vacancy.

That's it.

We take every guest in their room N, and ask them to move to room N+1, and now room 1 is vacant.

So the infinite hotel is always full but always has vacancy. That's what makes it a paradox.

There's already a cheap CO2 capture method, tried and tested: trees.

Ethanol has about 30% less energy per liter than gasoline. So a 10% ethanol:gasoline mix has about 3% less possible energy than 100% gasoline. That carries over into the mileage. and its worse in cold weather. Newer engines do use it better but you can't beat basic energy density.

40% of corn in the US is used for ethanol production. Its a shitton of other resources including land, fertilization, etc in order to do so.

https://mechbasic.com/does-ethanol-in-petrol-reduce-mileage/ https://www.newindianexpress.com/business/2025/Aug/05/govt-admits-lower-mileage-in-ethanol-blended-fuel

https://www.opensubsidies.org/analysis/ethanol-subsidy-machine https://www.taxpayer.net/energy-natural-resources/understanding-u-s-corn-ethanol-and-other-corn-based-biofuels-subsidies/

A black hole is a dead collapsed star with all its mass, that's why it has such a massive gravitational pull.

The turtle's dead, folks. Choked trying to vomit up another galaxy.

Ah-ha, I see. Thanks for that.

There are many types of event horizons

Really. That is intriguing!

With the interferometers they can estimate the mass of the 2 objects entering in collision and the resulting one.

Since there are multiple interferometers, they can also point to the region of the sky it happens. The "alert" can be picked up by other observatories which can look there on all frequencies.

Depending on if there is an explosion or not they can confirm if the objects were neutron stars or black hole (the details are a bit fuzzy on my head for the black hole / neutron star merger).

What if it's just turtles the whole way down?

Hawking radiation is not a confirmed effect and it is unique to event horizons not black holes. A naked singularity would have mass spin and charge but no hawking radiation. There are many types of event horizons and all of them have hawking radiation.

And also some measure of radiation, right? Hawking radiation? Whatever that is?

Yes the total electrical charge of the particles consumed is conserved

Huh. Didn't realize they had charge. As in electromagnetic charge?

You realize in a black hole like a teaspoon of whatever that is would probably have more mass than our entire planet? Maybe more than our sun.

There is no wide open spaces with gardens and birds and bees and oxygen to breathe.

Maybe moving from one to the other...but we're already inside.

Worth the watch.

I believe most of the time they run models with the known parameters and use that to decide what the objects were. It also seems that we appear to understand what different wave patterns look like and know what the source objects tend to be. Other times I believe they can figure out what one object was but not the other and just kinda have to guess at it. I would also suspect that once they determine the location in the sky where the event happened they can check and see if there are any past observations of that area of the sky by other telescopes and just look and see what was there previously.

I'm sure I'm greatly simplifying this, but that's what I turned up with a bit of reading on the LIGO website for Gravitational Wave Sources and Types.

Edit: typos

You would be spaghettified.

Black holes definitely have mass, which is calculated by the observations of motion of celestial bodies.

Indeed, Newton's 3rd law is 'equal and opposite forces.' so to stand up you need to exert a force upon the Earth at least equal to the force of it pulling upon you.

For instance, you Don, exert an extremely tiny gravitational 'force' upon the Earth and even the people around you and every object around you and even objects on the other side of the planet. Even the moon! And the planets and sun and so forth.

Newtons basic gravity equation defines force(F) equal to the gravitational constant (G) multipled by mass1(M) * mass2(m) divided by the radius(r) squared.

F = GMm/r²

Where G equals 6.67430 × 10^-11 m³ kg^-1 s^-2

Hope that helps.

This isn't really a mystery. Yes they all have mass. Yes they all pull on each other, as do all things with mass (which is basically all things, btw).

VWOOOP.

I can comprehend gravitational waves and the crazy precision needed to measure them, but I've still never been able to understand how they determine the objects that merge to cause the waves.

I see articles about having detected things like "two supermassive black holes", "a supermassive black hole and a neutron star", or "one massive black hole and an abnormally large poodle" colliding. I have no idea how they feel they have any real knowledge of what objects at what sizes led to the waves from simply being able to see the relative size of the waves and now long they last.