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tzimie

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From here: https://en.wikipedia.org/wiki/Measurable_cardinal

measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it can be described as a subdivision of all of its subsets into large and small sets such that κ itself is large, ∅ and all singletons {

*α*},*α*∈*κ*are small, complements of small sets are large and vice versa.**The intersection of fewer than***κ*large sets is again largeI don't understand the last part I've put in bold.

It is saying that

**k**can't be split into 2 "large" disjoint sets? For reals (not measurable cardinal of course), I can call a non-countable sets "large". Then x<0 and x>0 are both large with empty intersection. Why it won't work for measurable?

I can believe that measurable cardinal is so big, that there are so many elements, that (countable set of formulas) can't effectively discriminate then to build such sets. But the definition above say nothing about definability, but just about sets in general.And AC is so powerful that can build disjoint sets if we don't require formulas

Please help me understand it on intuitive level.