Iterating ordinal definability (Q791523)

This paper gives a large variety of results concerning HOD (the class of hereditarily ordinal definable sets) and its iterated versions \((HOD_{\alpha +1}=HOD^{HOD_{\alpha}};\quad HOD_{\lambda}=\cap_{\beta<\lambda}HOD_{\beta}).\) Some old results are stated and proved: this is used to present the key ideas and technical details (essentially McAloon's method of coding) of the new results. The old theorems are: \(\forall M \exists N\) generic extension s.t. \(HOD^ N=M\); \(\forall M \exists N\) generic extension s.t. \(N\vDash ZFC+V=HOD\) (Roguski); \(cons(ZF)\to cons(ZF+\forall n\quad HOD_ n\neq HOD_{n+1}\pm(AC)^{HOD_{\omega}})\) (McAloon). The new ones are: \(cons(ZF)\to cons(ZF+HOD_{\omega}\nvDash ZF)\) (unpublished from Harrington); \(\forall M \exists N\) generic extension s.t. \(HOD_{on}=M+\forall \alpha \quad HOD_{\alpha +1}\neq HOD_{\alpha}; \forall M\), \(\alpha\in M\), \(\exists N\) generic extension s.t. \((HOD_{\alpha +1})^ N=M+\forall \beta<\alpha \quad HOD_{\beta}\neq HOD_{\beta +1}.\) The paper is self contained. It is very well written and pleasant to read. A nice survey of the subject.