Independence results

hypothesis that no set has a cardinality between that of the integers and that of the real numbers

theorem

axiom in mathematical logic that all cardinals less than the cardinality of the continuum behave like ℵ₀ in a specific sense

the proposition, independent of ZFC, that a nonempty unbounded complete dense total order satisfying the countable chain condition is isomorphic to the reals

combinatorial principle that there exists a family of sets 𝐴(𝛼)⊆𝛼 for 𝛼<ω₁ such that for any 𝐴⊆ω₁, the set of 𝛼’s with 𝐴∩𝛼=𝐴(𝛼) is stationary in ω₁

certain principle in Ramsey theory is true, but not provable in Peano arithmetic

in set theory, a tree of uncountable height with no uncountable branches and no uncountable levels