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As soon as you start treating the axiom of choice as a superpower for a conscious being you are in conceptual la la land. How do I guess the color of my hat if
by Tloewald 10y ago
As soon as you start treating the axiom of choice as a superpower for a conscious being you are in conceptual la la land.
How do I guess the color of my hat if there's an uncountable number of possible colors? Doesn't that mean that communicating the value of that color involves transmitting an infinite amount of informtion?
- waqf 10y agoQuick, tell me your favourite number between 0 and 1. How did you do that? Weren't there an uncountable number of possible numbers? Ah, you may say, but I obviously wasn't going to choose one with an infinite information content, so all but countably many possible numbers had probability 0. Which is true. But in fact it's true that whenever you have a probability measure on an uncountable space then all but countably many elements have probability 0, so that escape clause applies equally well to the hat colours.
- TheCoelacanth 10y agoBut the axiom of choice applies to all sets, not just ones that you can easily choose a number from. For instance, what is your favorite non-computable number between 0 and 1?
- red75prime 10y agoInverse of Kolmogorov complexity of thirteenth bit-string with uncomputable Kolmogorov complexity, of course.
- Tloewald 10y agoWhen we think of numbers between 0 and 1 we aren't thinking of Real numbers. Name a transcendental number between 0 and 1. Which one springs to mind? Maybe, if you're lucky you can come up with some famous transcendental number and "scale" it to lie within the interval (pi over ten!). The numbers we live with are a finite dimensional vector field over a bounded subset of algebraics - not even countable.