These are cardinals big enough to have a 0,1 measure on their powerset.
Well ZFC can’t prove whether or not they exist. If you know what ultra-filters are, these are ulrafilters that meet the stronger condition of being closed under countable intersection, not just finite intersection.
There is the cantor distribution.
https://en.wikipedia.org/wiki/Cantor_distribution
One way of getting it is to take a coin, write 0 on one side and 2 on the other. Flip it infinity times. This gives you a number in trinary.
If you have a set A to measure, then μ(A)=P(c∈A) where c is the number made in trinary above.
There are also measurable cardinals. https://en.wikipedia.org/wiki/Measurable_cardinal
These are cardinals big enough to have a 0,1 measure on their powerset.
Well ZFC can’t prove whether or not they exist. If you know what ultra-filters are, these are ulrafilters that meet the stronger condition of being closed under countable intersection, not just finite intersection.