My thoughts have been pretty scattered and abstract lately. I have been thinking about History and some elementary but strange ideas about sets of numbers.

A Pfennig for a thought about history - What if they had a war and everybody did silly British theatre comedy dance routines?

I always wondered how history would have been different if the Axis forces in World War Two had a decent sense of rhythm and a sense of humor. Would that have given them the edge they lacked? Would North America have been divided up by the Axis powers, like in Philip K Dick's novel, “The Man in the High Castle”?

Today, I found evidence this may be a moot question. The Nazis had a great sense of humor and were masters of lighthearted comic dance routines. I base this completely on intelligence film provided by the British Ministry of Information.

Oh those Nutzy Nazis.

Slate magazine managed to dig this gem up. The British Department of Information (really propaganda) made a mash-up of “Triumph of the Will” to turn the Whole Munich Nazi-Fest into a giant production of the “Lambeth Walk”.

Infinite Pennies for a thought about Sets, Numbers and The Burger King at the End of the Universe.

The result – At the Burger King at the End of the Universe, patience is rewarded with free soda:

Everybody gets free soda after just one hasty person (the guy who cuts to the front of the line) pays! Provided they are polite and take orderly turns at the soda machine.

Two completely reasonable assumptions to make this easier:

Everybody stays there forever (because its the end of the universe),

You get access to the soda machine for unlimited free refills (because it's Burger King).

Oddly enough, the Mathematics that explains this situation uses ideas from Elementary School, although your fifth grade teacher probably didn't use these exactly this way:

Start out with just two customers, customer one and customer two.

Look at what Mathematicians call the Natural Numbers, just the numbers we use to count sheep;

(1,2,3,4,5, and so on) This is the set of natural numbers.

Multiply each of these numbers by 2, and you get the set of even numbers; (2,4,6,8,10, and so on).

You can match each natural number to a unique even number, and each even number to a unique natural number, like this:

1<-->2

2<-->4

3<-->6

and so on.

SO WHAT?

Because you can pair every natural number uniquely with an even number, You can say there are no more or fewer even numbers than natural numbers. The two sets of numbers are the same size.

BUT WHAT ABOUT BURGER KING?, you may ask.

Let's count the cups poured from the soda machine.

The first guy in line has to pay for sodas equal to the size of the natural numbers, because he'll drink forever.

The second (and more polite guy) can make a case to get his soda for free. After all, the first guy has already paid for an infinite number of sodas. So there. And, the first guy will still get all his sodas.

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This is an elementary bit of Set Theory and a basic idea underlying most Mathematical advancement since the mid-19^th Century. Georg Cantor developed these ideas and took a lot of heat because the consequences of these ideas caused decades of upheaval among European Theologians for reasons that seem trivial to me. To this day, some religious extremists remain suspicious of set theory in school Math instruction.

If your hypothetical customers aren't hungry, they can check in to the Hilbert Grand Hotel and get the same strangeness. It never ends - literally.