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See the continuum hypothesis. The cardinality of the real numbers is two raised to the power of the cardinality of the natural numbers. So: given that there are
by xigency 12y ago
See the continuum hypothesis. The cardinality of the real numbers is two raised to the power of the cardinality of the natural numbers. So: given that there are a countably infinite number of divisions, each doubling the pixel count, there must be uncountably many pixels that result.
This is more obvious if you simply view the "infinite screen" as a bounded region of space with coordinates denoted by pairs of real numbers.
- xrayspec 12y agoOrdinal 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#Exponentiation http://en.wikipedia.org/wiki/Ordinal_arithmetic#Exponentiati...
- xigency 12y agoAnd 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:Algebraicszoom.png http://en.wikipedia.org/wiki/Algebraic_number#/media/File:Al...
- deleted 12y ago[deleted]