Halpern–Läuchli theorem

Halpern–Läuchli theorem In mathematics, the Halpern–Läuchli theorem is a partition result about finite products of infinite trees. Its original purpose was to give a model for set theory in which the Boolean prime ideal theorem is true but the axiom of choice is false. It is often called the Halpern–Läuchli theorem, but the proper attribution for the theorem as it is formulated below is to Halpern–Läuchli–Laver–Pincus or HLLP (named after James D. Halpern, Hans Läuchli, Richard Laver, and David Pincus), following Milliken (1979).

Let d,r < ω, {displaystyle langle T_{i}:iin drangle } be a sequence of finitely splitting trees of height ω. Let {displaystyle bigcup _{nin omega }left(prod _{i

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