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Constant comments on The Many Worlds of Hugh Everett - Less Wrong
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Comments (45)
The set of all clock faces a working clock can produce - call this the set of all valid clock faces - has the same topology (and cardinality) as a circle. The set of all possible clock faces has the same topology (and cardinality) as a 2-dimensional torus.
However, the cardinality of a 2-dimensional torus is the same as the cardinality of a square, which is the same as the cardinality of a line (as you yourself recognize), which is the same as the cardinality of a circle.
Therefore the set of all valid clock faces has the same cardinality as the set of all possible clock faces.
A power set indeed has a larger cardinality than the set it is a power set of. However, the set of all possible clock faces is not the power set of the set of all valid clock faces.
Yes.
Show me.
John K Clark
There are two hands, an hour hand and a minute hand. The set of all possible positions that the hour hand can take describes a circle. The same is true of the minute hand: its set of all possible positions describes a circle. Consequently, the set of all ordered pairs of possible positions (h,m), where h is the position of the hour hand and m is the position of the minute hand, is the Cartesian product of the two individual sets, and thus the Cartesian product of two circles. This is a two-dimensional torus.
Did you take into account that the positions of the two hands are not independent? When the hour hand of a given clock is at its 12:00:00 position, there's only one possible location of the minute hand for that clock, and this is true for any position of the hour hand.
If you read elsewhere in the thread, you'll see that johnclark draws a distinction between all possible clock faces and "valid clock faces", i.e., those that obey the constraint you describe. Constant is addressing the former, not the latter.
Yes. Thanks.
Ah. Okay.
Good point. The minute hand is entirely redundant.