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# failure of \square(\kappa)$\square(\kappa)$ at an inaccessible \kappa$\kappa$

How can we force the failure of \square(\kappa) $\square(\kappa)$ at an inaccessible \kappa, $\kappa$, where \square(\kappa) $\square(\kappa)$ is defined as follows: There is a sequence (C_i:i$(C_i:i< \kappa) kappa)$ such that:

(1) C_{i+1} $C_{i+1} = {i} \{i\}$ and C_i $C_i$ is closed and cofinal in i $i$ if i $i$ is a limit ordinal.

(2) If i $i$ is a limit point of C_j, $C_j$, then $C_i = C_j \cap ii$.

(3) There is no club C $C$ (a subset of \kappa) $\kappa$) such that for all limit points i $i$ in C $C$ the equality $C_i= C \cap i i$ holds.

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# failure of \square(\kappa) at an inaccessible \kappa

How can we force the failure of \square(\kappa) at an inaccessible \kappa, where \square(\kappa) is defined as follows: There is a sequence (C_i:i< \kappa) such that: (1) C_{i+1} = {i} and C_i is closed and cofinal in i if i is a limit ordinal. (2) If i is a limit point of C_j, then C_i = C_j \cap i. (3) There is no club C (a subset of \kappa) such that for all limit points i in C the equality C_i= C \cap i holds.