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  "body": "\n## Summary\n\nA weakly compact cardinal is an uncountable cardinal $\\kappa$ for which the infinite Ramsey theorem holds at $\\kappa$.\nThis means that every colouring of the pairs of elements of $\\kappa$ with two colours has a homogeneous set of size $\\kappa$.\nWeakly compact cardinals are [[inaccessible|NW4DREQ5DQ]] and even [[Mahlo|CJFEPPP9KA]].\nThe name comes from an equivalent property: a compactness theorem for the infinitary logic $\\mathcal{L}_{\\kappa, \\kappa}$ in the case of theories with at most $\\kappa$ sentences.\nEvery [[measurable cardinal|AKBWT323RQ]] is weakly compact.\n\n## Explanations\n\n### L1 human\n\nRamsey's theorem says that if one colours each pair of natural numbers red or blue, there is an infinite set of numbers whose pairs all have the same colour.\nA weakly compact cardinal is an uncountable size of infinity for which the same is true.\nFor the first uncountable cardinal this fails, so a weakly compact cardinal must be very large.\nIt is a large cardinal that combines combinatorics, trees and logic in one notion.\n",
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