No. Each clock face has the same cardinality as the points of a line, but the set of possible clock faces (which I take to mean something like “ways of colouring a disc”, where how many colours you’re allowed doesn’t really matter) has the same cardinality as the set of ways of colouring a line, or equivalently the set of subsets of the line, in other words 2^continuum.
Voted down for starting off with “no” before it was established what what was being talked about. This all depends on what he meant by “clock face”. Until that’s clear it’s simply impossible to say.
I’d been going to reply and say “butbutbutbut the only possible interpretation of ‘clock face’ that would make the cardinality not be 2^c would be that ‘clock face’ means ‘pair of clock-hand positions’, and that’s just ridiculous” … and then I looked at the rest of the thread and saw that that was apparently what JKC meant. My apologies.
EDITED to add: er, and actually there are perfectly reasonable interpretations that make the cardinality only the same as that of R, such as anything that can be described completely by a finite set of simple closed curves, or an RGB colouring where the colour at each point is a continuous function of position. So: total fail. Sorry again.
No. Each clock face has the same cardinality as the points of a line, but the set of possible clock faces (which I take to mean something like “ways of colouring a disc”, where how many colours you’re allowed doesn’t really matter) has the same cardinality as the set of ways of colouring a line, or equivalently the set of subsets of the line, in other words 2^continuum.
Voted down for starting off with “no” before it was established what what was being talked about. This all depends on what he meant by “clock face”. Until that’s clear it’s simply impossible to say.
I’d been going to reply and say “butbutbutbut the only possible interpretation of ‘clock face’ that would make the cardinality not be 2^c would be that ‘clock face’ means ‘pair of clock-hand positions’, and that’s just ridiculous” … and then I looked at the rest of the thread and saw that that was apparently what JKC meant. My apologies.
EDITED to add: er, and actually there are perfectly reasonable interpretations that make the cardinality only the same as that of R, such as anything that can be described completely by a finite set of simple closed curves, or an RGB colouring where the colour at each point is a continuous function of position. So: total fail. Sorry again.