[HTML][HTML] Characterizing large cardinals in terms of layered posets

S Cox, P Lücke - Annals of Pure and Applied Logic, 2017 - Elsevier
Annals of Pure and Applied Logic, 2017Elsevier
Given an uncountable regular cardinal κ, a partial order is κ-stationarily layered if the
collection of regular suborders of P of cardinality less than κ is stationary in P κ (P). We show
that weak compactness can be characterized by this property of partial orders by proving
that an uncountable regular cardinal κ is weakly compact if and only if every partial order
satisfying the κ-chain condition is κ-stationarily layered. We prove a similar result for strongly
inaccessible cardinals. Moreover, we show that the statement that all κ-Knaster partial …
Given an uncountable regular cardinal κ, a partial order is κ-stationarily layered if the collection of regular suborders of P of cardinality less than κ is stationary in P κ (P). We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal κ is weakly compact if and only if every partial order satisfying the κ-chain condition is κ-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all κ-Knaster partial orders are κ-stationarily layered implies that κ is a Mahlo cardinal and every stationary subset of κ reflects. This shows that this statement characterizes weak compactness in canonical inner models. In contrast, we show that it is also consistent that this statement holds at a non-weakly compact cardinal.
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