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The set of pixels along the edge of your choice is countable. The reason is basically the same as was given, except it's simpler since you're only subdividing i
by roywiggins 12y ago
The set of pixels along the edge of your choice is countable. The reason is basically the same as was given, except it's simpler since you're only subdividing in one dimension. You'll never drop a pixel exactly at 1/3 or anywhere that isn't an integer multiple of a power of 2.
You can identify each pixel by two other pixels along your chosen edges (axes). The set of pairs drawn from countable sets is always countable.
- pdonis 12y ago> The set of pixels along the edge of your choice is countable. It is for a finite number of divisions. But is it for an infinite number of divisions? You are basically assuming that the limit point of a given sequence must have the same properties as every item in the sequence. That is obviously false; for example, many limit points of sequences of rational numbers are not rational numbers. So you can't just help yourself to the assumption that, because each pixel's coordinates have a certain property after a finite number of divisions, the coordinates will still have the same property after an infinite number of divisions.
- Dylan16807 12y agoEach pixel always starts at coordinate n / 2^k. There are no limits involved here. Just a construction of an infinite series of rational coordinates.