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Ali Enayat
Ali Enayat
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Two comments/answers:

(1) By an old theorem of Roguski, for any $\Sigma_2^{\text{ZFC}}$ sentence $\phi$, the theories $\text{ZFC} + \phi$ and $\text{ZFC + V=HOD} + \phi$ are equiconsistent.

Roguski's result appears in his paper Extensions of models for ZFC to models for ZF+V=HOD with applications, in Set theory and hierarchy theory, pp. 241–247. Lecture Notes in Math., Vol. 537, Springer, Berlin, 1976.

(2) Since there is a parenthetical reference to fragments of $\text{ZF}$ in the first line of the question: the formulation and salient consequences of $\text{V=HOD}$ heavily depend on stratification of the universe into rank initial segments of the form $V_\alpha$, and on the veracity of the Montague-Levy reflection theorem; the latter is equivalent over $\text{ZF}$ without $\{\text{Replacement, Infinity}\}$ to the conjunction of $\text{Replacement}$ and $\text{Infinity}$ (by an old result of Azriel Levy).

So $\text{V=L}$, rather than $\text{V=HOD}$, is the safe way to arrange global choice, at least for fragments of $\text{ZF}$ that are extend Kripke-Platek set theory.

Two comments/answers:

(1) By an old theorem of Roguski, for any $\Sigma_2^{\text{ZFC}}$ sentence $\phi$, the theories $\text{ZFC} + \phi$ and $\text{ZFC + V=HOD} + \phi$ are equiconsistent.

Roguski's result appears in his paper Extensions of models for ZFC to models for ZF+V=HOD with applications, in Set theory and hierarchy theory, pp. 241–247. Lecture Notes in Math., Vol. 537, Springer, Berlin, 1976.

(2) Since there is a parenthetical reference to fragments of $\text{ZF}$ in the first line of the question: the formulation and salient consequences of $\text{V=HOD}$ heavily depend on stratification of the universe into rank initial segments of the form $V_\alpha$, and on the veracity of the Montague-Levy reflection theorem; the latter is equivalent over $\text{ZF}$ without $\{\text{Replacement, Infinity}\}$ to the conjunction of $\text{Replacement}$ and $\text{Infinity}$ (by an old result of Azriel Levy).

So $\text{V=L}$, rather than $\text{V=HOD}$, is the safe way to arrange global choice, at least for fragments of $\text{ZF}$ that are extend Kripke-Platek set theory.

Two comments/answers:

(1) By an old theorem of Roguski, for any $\Sigma_2^{\text{ZFC}}$ sentence $\phi$, the theories $\text{ZFC} + \phi$ and $\text{ZFC + V=HOD} + \phi$ are equiconsistent.

Roguski's result appears in his paper Extensions of models for ZFC to models for ZF+V=HOD with applications, in Set theory and hierarchy theory, pp. 241–247. Lecture Notes in Math., Vol. 537, Springer, Berlin, 1976.

(2) Since there is a parenthetical reference to fragments of $\text{ZF}$ in the first line of the question: the formulation and salient consequences of $\text{V=HOD}$ heavily depend on stratification of the universe into rank initial segments of the form $V_\alpha$, and on the veracity of the Montague-Levy reflection theorem; the latter is equivalent over $\text{ZF}$ without $\{\text{Replacement, Infinity}\}$ to the conjunction of $\text{Replacement}$ and $\text{Infinity}$ (by an old result of Azriel Levy).

So $\text{V=L}$, rather than $\text{V=HOD}$, is the safe way to arrange global choice, at least for fragments of $\text{ZF}$ that extend Kripke-Platek set theory.

Source Link
Ali Enayat
Ali Enayat
  • 17.7k
  • 2
  • 63
  • 105

Two comments/answers:

(1) By an old theorem of Roguski, for any $\Sigma_2^{\text{ZFC}}$ sentence $\phi$, the theories $\text{ZFC} + \phi$ and $\text{ZFC + V=HOD} + \phi$ are equiconsistent.

Roguski's result appears in his paper Extensions of models for ZFC to models for ZF+V=HOD with applications, in Set theory and hierarchy theory, pp. 241–247. Lecture Notes in Math., Vol. 537, Springer, Berlin, 1976.

(2) Since there is a parenthetical reference to fragments of $\text{ZF}$ in the first line of the question: the formulation and salient consequences of $\text{V=HOD}$ heavily depend on stratification of the universe into rank initial segments of the form $V_\alpha$, and on the veracity of the Montague-Levy reflection theorem; the latter is equivalent over $\text{ZF}$ without $\{\text{Replacement, Infinity}\}$ to the conjunction of $\text{Replacement}$ and $\text{Infinity}$ (by an old result of Azriel Levy).

So $\text{V=L}$, rather than $\text{V=HOD}$, is the safe way to arrange global choice, at least for fragments of $\text{ZF}$ that are extend Kripke-Platek set theory.