Originally Posted by

**TridentBlue**
You are confused between a set and its elements. Why can't I have a finite or countable set of elements; where each element is in fact an uncountable set or an incompressible real?

I hear you, and you totally can. How about the singleton set with the set of real numbers mapped to 1? I may have used sloppy language in making my point, but my point is good: The basics of what I'm saying is that if you define some mapping from members of some infinite set to the natural numbers, and that mapping results in some clearly defined member of your set mapping to a natural number with infinite digits, (infinite information) than that set is not countable by that mapping.

For instance, suppose I claim that by mapping the decimals between 0 and 1 to the the naturals, as I did above, (so 0.1 =1, 0.55 = 55, 0.12334 = 12,334, etc) that I have found a way to count the rational numbers. You come along and point out as counter example 1/3, which is a rational but maps to the natural number 333333.... a natural number with infinite digits, all 3's. That means that my mapping does not count the rational numbers. Note that this does not mean the rational numbers are uncountable, as we both know they are. Just that my mapping fails to count them all.

A set of numbers is uncountable when no mapping to the natural numbers exists which doesn't have SOME members of the set requiring infinite-digit-naturals to represent them, aka infinite information. Trivially, every finite set is countable. If we allow that non-algorithmically compressible numbers can be well defined (I'm not sure about this) than this doesn't change the fundamentals: Even if it takes infinite information to represent the number,

**it must not take infinite information to represent the natural number it maps to** : it cannot map to a natural number with infinite digits. You need to always be able to come up with a map that will, in all cases, represent this number with a finite amount of information. With uncountable sets, this is simply impossible.... Its not impossible to come up with a map that represents some numbers which are infinitely long with vast information, but its impossible to come up with a map that represents ALL such numbers.