Hilbert’s Paradox of the Grand Hotel

Post 348

Imagine a Grand Hotel. The plushest decor adorning the finest façade of handmade pale beige brick, the windows framed in a dark, rich red brick surround. There is beautiful ornate handrails and wonderfully crafted stonework. Topped off with a roof of supreme grace. Pure Victorian elegance. But this is no ordinary hotel. Inside, there are an infinite number of rooms, each filled with an infinite number of guests. A lowly traveller, weary from a long and arduous pilgrimage, arrives. The receptionist informs him that there are no rooms. But wait! There are an infinite number of rooms! Simply move the guest in Room 1 down to Room 2, and the guest in Room 2 down to Room 3, and so on. But, says the traveller, wouldn’t that create two sets of infinity? If the infinite rooms are filled with infinite people, then aren’t they, in reality, not infinite at all? What is infinity? ‘Tis the question, readers.

David Hilbert was a German mathematician. He spent his life in the field, and in the 1920s, he came up with this paradox. He only lived circa 20 years after this thought. He had no idea the millions of headaches he would create and the can of worms he not only opened, but also threw in our faces. He asked the most intriguing of questions. Is there such a thing as infinity? Well, you may be thinking, who cares? Probably isn’t. Does it matter? The question I would ask is, who governs space? If everything is finite, then it is controlled. It’s predetermined. But who or what predetermines it? Hmm? Do we have proof of infinity? If we don’t, this paradox is void. But what about the universe? It’s expanding. Science tells us it can expand an infinite number of times. Infinity does exist. Yet this paradox tells us it can’t.

One could argue that we could just keep moving the guests, continually, but that isn’t the point. Infinite rooms, infinite guests. It just isn’t possible. You cannot even argue that there are levels of infinity, because Infinity A intersects with Infinity B to the point that rules upon layers of rules need to be created, and that’s not the point either. Infinity is black and white. It is an unending number of things. Period.

So what is the solution? Simple, really. Infinity is an invented construct in any case. Humankind invented that word and that concept. The universe doesn’t answer to us, we answer to it. We may have a concept of infinity, but we do not know, and cannot know, if the universe is infinite, or what the rules of infinity it abides by, are. There is a solution. But the reason we don’t know what that solution is, is because everything we know, we created. And we haven’t created an answer for it yet. Which is the answer to this paradox. Simple.

As for our weary traveller, well, what happened to him is a simple resolution, too. The Grand Hotel was well known. The hotel of infinity. It pissed the hell out of a lot of people, so those people created their own hotels around it, ordinary, dull and finite. Our weary traveller simply left The Grand Hotel and booked a room in the hotel over the road.

Hilbert never thought of that, did he?

How would you solve this paradox, readers?

Ciao :)(:


Photo: The Grand Hotel in Scarborough, England [not the one in the question, it just so happens that there is a hotel near where I live called The Grand Hotel]


I’d love to hear your thoughts on this post. You can leave a comment and/or like this post below, or by clicking the little bubble on the top right if you are on the Archives Page. Likes and follows greatly appreciated. Thanks.


Please feel free check out the latest posts from my other two blogs:

The Indelible Life of Me
New Post Every Sunday
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Hark Around the Words
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