[xrayspec]

Ordinal and cardinal exponentiation use similar notation, but they don't work the same way. Though it's common to see statements like "ω is the same as \aleph_0," it's misleading, because 2^ω = ω, and thus has lesser cardinality than 2^\aleph_0.

See also the "warning" here:

http://en.wikipedia.org/wiki/Ordinal_arithmetic#Exponentiati...

[xigency]

And here we have issues of ordering and uniqueness. There are multiple sequences that correspond to any one point, and in the countably infinite construction, there are points that have no sequence. This is not to say that the first definition is invalid or that the second definition is invalid.

It is possible to construct an infinite set that does not contain certain numbers. For example, if the coordinate (1/pi, 1/2) were excluded.

In some ways, what is being described at the end is, instead of a fully-filled screen, something that looks like this: http://en.wikipedia.org/wiki/Algebraic_number#/media/File:Al...