A full hotel with infinite rooms can still fit in infinitely many more guests

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A full hotel with infinite rooms can still fit in infinitely many more guests

the gist

Hilbert's paradox of the Grand Hotel imagines a hotel with infinitely many rooms that is completely full. When a new guest arrives, the manager moves the guest in room 1 to room 2, the guest in room 2 to room 3, and so on, freeing room 1. Even infinitely many new guests can be fitted in by moving each current guest to the room with double their number, leaving all the odd rooms empty. The point is that infinite sets behave differently from finite ones: a set can be matched one-to-one with a part of itself. David Hilbert introduced the idea in a 1924 lecture.

how it works

Infinite sets can be matched one-to-one with proper subsets of themselves. Counting the rooms 1, 2, 3 and so on, every guest can shift to the next number, which always leaves a room free at the start. Moving guests to even-numbered rooms creates an infinite set of empty rooms for infinitely many newcomers.

so what

The paradox captures a central idea of set theory: some infinities are the same size as their own subsets, which overturns everyday intuition about counting. It helped make abstract ideas about infinity accessible to a general audience.

Hampi was once one of the world's largest cities, until its Vijayanagara capital was destroyed in 1565

the gist

Hampi is a UNESCO World Heritage Site in Karnataka, India, with more than 1,600 surviving remains spread over 4,100 hectares. It was the capital of the Vijayanagara Empire from 1336 to 1565. Travellers from Persia and Portugal described it as a prosperous, wealthy and grand city with temples, markets and farms along the Tungabhadra River. By 1500 it was estimated to be the world's second-largest city after Beijing, and probably India's richest at the time. In 1565 a coalition of Muslim sultanates defeated the Vijayanagara Empire, its capital was conquered and destroyed, and the city was left in ruins. Hampi remains a religious centre, with the active Virupaksha Temple at its heart.

how it works

The city was built around a fortified core with temples, markets and water structures along the river, supplied by an extensive network of roads and waterworks. When the empire fell, the city was not rebuilt, so its stone monuments were left standing.

so what

Hampi shows how quickly a great city can vanish after a military defeat, and how a ruined capital can remain sacred. Its ruins give historians a view of medieval South Asia that few other sites can match.

fun fact

The name Hampi comes from Pampa, an old name for the Tungabhadra River.

From Wikipedia, the free encyclopedia · text CC BY-SA 4.0

Image: Wikimedia Commons

source: Wikipedia ↗

2 min

politicsshortCNA2:01

False allegations about Singapore’s hand in Indonesia’s markets and a made-up meeting

from CNA

CNA Verify has found close to 400 posts pushing falsehoods tying Singapore to Indonesia’s stock market rout and to a secret meeting that never took place. Together, they drew nearly 12 million views, and some were circulated by accounts that appeared to be working in concert, across multiple social media platforms.

Here’s how the disinformation campaign targeting Singapore was uncovered.

on YouTube: CNA ↗