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edited Jan 1, 2016 at 19:08

On one hand due to Kunen's inconsistency theorem it is known that within $\sf ZF$, large cardinal axioms beyond Reinhardt cardinal are inconsistent with $\sf AC$.

Also some recent results of Bagaria, Koellner and Woodin (see here) suggest that very large cardinals beyond Reinhardt (e.g. Berkeley cardinals), could be inconsistent with weaker choice principles like $\sf DC$ under some "plausible assumptions" (borrowed from Koellner's words in his lecture slides).

Now consider the Konig's Infinity Lemma which implies tree property at $\aleph_0$, the statement that every $\aleph_0$ - tree has a cofinal branch. It is not hard to see that there is a similarity between such cofinal branches in $\kappa$ - trees and the chains$R$-chains that $\sf DC$ guarantees their existenceproduces for a binary relation $R$ on a set $X$. In the other words tree property could be considered as a kind of Axiom of Dependent choiceChoice. (For more information see here).

Now by replacing $\sf DC$ with tree property at Bagaria, Koellner and Woodin's observation, it is natural to ask:

Question. Within $\sf ZF$, is there any inconsistency between very large cardinal axioms beyond Reinhardt cardinals and tree property at one or more regular cardinals?

On one hand due to Kunen's inconsistency theorem it is known that within $\sf ZF$, large cardinal axioms beyond Reinhardt cardinal are inconsistent with $\sf AC$.

Now consider the Konig's Infinity Lemma which implies tree property at $\aleph_0$, the statement that every $\aleph_0$ - tree has a cofinal branch. It is not hard to see that there is a similarity between such cofinal branches in $\kappa$ - trees and the chains that $\sf DC$ guarantees their existence. In the other words tree property could be considered as a kind of Axiom of Dependent choice. (For more information see here).

Now it is natural to ask:

Question. Within $\sf ZF$, is there any inconsistency between very large cardinal axioms beyond Reinhardt cardinals and tree property at one or more regular cardinals?

On one hand due to Kunen's inconsistency theorem it is known that within $\sf ZF$, large cardinal axioms beyond Reinhardt cardinal are inconsistent with $\sf AC$.

Now consider the Konig's Infinity Lemma which implies tree property at $\aleph_0$, the statement that every $\aleph_0$ - tree has a cofinal branch. It is not hard to see that there is a similarity between such cofinal branches in $\kappa$ - trees and the $R$-chains that $\sf DC$ produces for a binary relation $R$ on a set $X$. In the other words tree property could be considered as a kind of Axiom of Dependent Choice. (For more information see here).

Now by replacing $\sf DC$ with tree property at Bagaria, Koellner and Woodin's observation, it is natural to ask:

Question. Within $\sf ZF$, is there any inconsistency between very large cardinal axioms beyond Reinhardt cardinals and tree property at one or more regular cardinals?

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asked Jan 1, 2016 at 19:02

Morteza Azad

Is tree property inconsistent with Berkeley cardinals in the absence of Axiom of Choice?

On one hand due to Kunen's inconsistency theorem it is known that within $\sf ZF$, large cardinal axioms beyond Reinhardt cardinal are inconsistent with $\sf AC$.

Now consider the Konig's Infinity Lemma which implies tree property at $\aleph_0$, the statement that every $\aleph_0$ - tree has a cofinal branch. It is not hard to see that there is a similarity between such cofinal branches in $\kappa$ - trees and the chains that $\sf DC$ guarantees their existence. In the other words tree property could be considered as a kind of Axiom of Dependent choice. (For more information see here).

Now it is natural to ask:

Question. Within $\sf ZF$, is there any inconsistency between very large cardinal axioms beyond Reinhardt cardinals and tree property at one or more regular cardinals?