马修.埃蒙斯 视频:希尔伯特旅馆悖论（1）

Today is the 100th day of school. He was very excited and told me everything he knows about the number 100, including that 100 was an even number. He then told me that 101 was an odd number and 1 million was an even number, etc.  He then paused and asked: “Is infinity even or odd?”

I explained that infinity is neither even nor odd.  It’s not a number in the usual sense, and it doesn’t obey the rules of arithmetic.  All sorts of contradictions would follow if it did.  For instance, “if infinity were odd, 2 times infinity would be even.  But both are infinity!  So the whole idea of odd and even does not make sense for infinity.”

Kim replied:

Thank you.  Ben was satisfied with that answer and kind of likes the idea that infinity is big enough to be both odd and even.

Although something got garbled in translation (infinity is neither odd nor even, not both), Ben’s rendering hints at a larger truth.  Infinity can be mind-boggling.

Some of its strangest aspects first came to light in the late 1800s, with Georg Cantor’s groundbreaking work on “set theory.”  Cantor was particularly interested in infinite sets of numbers and points, like the set {1, 2, 3, 4,…} of “natural numbers” and the set of points on a line.  He defined a rigorous way to compare different infinite sets and discovered, shockingly, that some infinities are bigger than others.

At the time, Cantor’s theory provoked not just resistance, but outrage.  Henri Poincaré, one of the leading mathematicians of the day, called it a “disease.”  But another giant of the era, David Hilbert, saw it as a lasting contribution and later proclaimed, “No one shall expel us from the Paradise that Cantor has created.”

My goal here is to give you a glimpse of this paradise.  But rather than working directly with sets of numbers or points, let me follow an approach introduced by Hilbert himself.  He vividly conveyed the strangeness and wonder of Cantor’s theory by telling a parable about a grand hotel, now known as the Hilbert Hotel.

It’s always booked solid, yet there’s always a vacancy.

旅馆永远客满，却永远有空房间接待新的客人。