Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given an uncountable regular cardinal  κ , a partial order is  κ   -stationarily layered if the collection of regular suborders of   P     P        of cardinality less than  κ    is stationary in   P κ (P)       P    κ    (  P  )       .  show that ak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal  κ  is akly compact if and only if every partial order satisfying the  κ -chain condition is  κ -stationarily layered.  prove a similar result for strongly inaccessible cardinals. Moreover,  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 ak compactness in canonical inner models. In contrast,  show that it is also consistent that this statement holds at a non-akly compact cardinal.
['Sean Cox', 'Philipp Lücke']
I am writing a research paper on a idea Characterizing large cardinals in terms of layered posets
[{"from": "human", "value": "I am writing a research paper on a idea Characterizing large cardinals in terms of layered posets"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given an uncountable regular cardinal  \u03ba , a partial order is  \u03ba \u00a0 -stationarily layered if the collection of regular suborders of   P     P        of cardinality less than  \u03ba \u00a0  is stationary in   P \u03ba (P)       P    \u03ba    (  P  )       .  show that ak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal  \u03ba  is akly compact if and only if every partial order satisfying the  \u03ba -chain condition is  \u03ba -stationarily layered.  prove a similar result for strongly inaccessible cardinals. Moreover,  show that the statement that all  \u03ba -Knaster partial orders are  \u03ba -stationarily layered implies that  \u03ba  is a Mahlo cardinal and every stationary subset of  \u03ba  reflects. This shows that this statement characterizes ak compactness in canonical inner models. In contrast,  show that it is also consistent that this statement holds at a non-akly compact cardinal."}]