The real numbersYou might think it's easy to list the real numbers. Start at 0, and then you have 0.1, 0.2, 0.3... But wait, between 0 and 0.1 you have 0.01, 0.02 and so on. And between 0 and 0.01 you have 0.001, 0.002... In fact, it is impossible to get anywhere at all, because there is always a finer-grained way of listing the numbers. Unlike the natural numbers (positive integers), which you can simply list as 1, 2, 3... Georg Cantor proved that this was the case. And that's why the real numbers are in a sense more infinite than the natural numbers.
Cantpr later tried to prove that nothing could at the same time be more infinite than the natural numbers and less infinite than the real numbers. He couldn't, so he went crazy and ended up in a mental asylum.
The list of all lists that do not contain themselvesSure, there are many lists that do not contain themselves. Your laundry list is presumably one of those, even if you write the list on a piece of paper you forget in your jeans (you'd only be washing the paper, not the list). An example of a list that does contain itself is the list of all lists of lists. And of course the list of all lists also contains itself. But the question is whether the list of all lists that do not contain themselves should list itself or not. Because if it does, it shouldn't, and if it doesn't, it should. Bertrand Russell spent a number of years puzzling over this so you don't have to.
This was important, because lists of lists are necessary in order to show that mathematics are true. Having thus destroyed attempts at founding mathematics on logic, Russell went on to be imprisoned for pacifism.
|Thinking about infinity might make you want to hide.|
All the animalsJorge Luis Borges provided the following list of animals, in a fictional text about a text:
- Those that belong to the emperor
- Embalmed ones
- Those that are trained
- Suckling pigs
- Mermaids (or Sirens)
- Fabulous ones
- Stray dogs
- Those that are included in this classification
- Those that tremble as if they were mad
- Innumerable ones
- Those drawn with a very fine camel hair brush
- Et cetera
- Those that have just broken the flower vase
- Those that, at a distance, resemble flies
Borges later lost his eyesight, but apparently never his virginity.
|There are some pretty weird animals out there.|
All statements that are true in a formal system of sufficient powerKurt Gödel thought a lot about truth. In particular, about which statements were true and which were not. To find out which statements were true, he invented a way to list all the statements in a system. Even more impressively, using this list he could figure out whether the statement was true just by looking at the number of it. Because the statements could be about anything within the system, there must be a statement in the list which talks about whether this statement itself is true - a statement that contains the proof of itself. As with all other statements, the number of this statement says whether it is true or false. Gödel showed that this statement is false. This means that is is impossible to list all true statements in a formal system, or all false statements for that matter. Rather disheartening really if you want to believe that the truth is in there.
Gödel, always a somewhat troubled fellow, later starved himself to death.
All steps you take as you follow a coastline
|Following the coastline of Britain in larger and smaller steps. Sorry, no animals. Image from Wikipedia.|
Mandelbrot escaped the nazis and went on to live a long (how long?) and apparently happy life.