0.999... Repeating Is Equal To 1, But Something Like It Is Not (Introduction To The Surreal Numbers)

Some of the comments appear to be based solely on reading the title or the thumbnail--so I've changed the title to more accurately reflect the contents. The surreal numbers contain the real numbers, and 0.9 repeating is equal to 1 in both. However, we think about 0.9 repeating conceptually as some number that is infinitely smaller than 1. There is such a number in the surreal numbers, and it is not equal to 1!

Technical Points
The surreal numbers are a pretty complicated logical system. One point to make is that the surreal numbers contain the real numbers, and so 0.999.. = 1 since this is true for the real numbers.

However, there is a psychology about 0.999... repeating as a number that is infinitesmally smaller than 1. The surreal number {0, 1/2, 3/4, 7/8, ... | 1} is infinitesmally smaller than 1 and would satisfy our psychological idea of 0.999.. repeating. I believe {0, 9/10, 99/100, 999/1000, ...| 1} would also represent the same surreal number; however, I have not been able to verify that so I did not include it in the video.

When fact-checking this video, I came across a number of conflicting arguments that seemed valid.

http://math.stackexchange.com/questio...
http://forums.xkcd.com/viewtopic.php?...
http://en.wikipedia.org/wiki/0.999......

I also did my best to work through John Conway's book that introduces surreal numbers called "On Numbers and Games."
http://en.wikipedia.org/wiki/On_Numbe...

I welcome thoughtful comments about the surreal numbers.

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