It is known that locally one can ``code'' any set in the von Neumann universe $V$ by a set of ordinals. But can one do this globally? In other words, is there a first-order definable bijection $P(On) \longrightarrow V$, where $P(On)$ is the class of all subsets of the class $On$ of all ordinals?


Note that $P(On)$ is in natural bijection with the ordered field $No$ of surreal numbers. Thus, the question is equivalent to: Is there a first-order definable bijection $No \longrightarrow V$?


If we assume $V = HOD$, then indeed there a first-order definable bijection $No \longrightarrow V$ (since $V = HOD$ is equivalent to the existence of a first-order definable bijection $On \longrightarrow No$ (or $On \longrightarrow V$); see [Definable map from all the ordinals to the surreal numbers with a dense image?](https://mathoverflow.net/questions/93468/definable-map-from-all-the-ordinals-to-the-surreal-numbers-with-a-dense-image/93485?noredirect=1#comment473182_93485)) However, I would be surprised if the converse were also true. It seems like the existence of a first-order definable bijection $No \longrightarrow V$ (or $P(On) \longrightarrow V$) should be weaker than $V = HOD$ but still perhaps not provable in ZFC. 


EDIT: Given Andreas Blass' answer that the statement is indeed independent of ZFC, I'm still wondering whether or not any of the implications $(1) \Rightarrow (2) \Rightarrow (3)$ are reversible, where the respective statements are (1) $V = HOD$, (2) there is a definable bijection $P(On) \longrightarrow V$, and (3) there is a definable linear ordering of $V$.



First-order definable bijection between $P(On)$ (or $No$) and $V$? (Is this equivalent to $V = HOD$?)

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> I will use $(1)$, $(2)$, and $(3)$ as defined in the EDIT to the question.
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> It is known that $(1)\Rightarrow (2)$ does not reverse over ZF, indeed, by a result due to Solovay (appearing as Theorem 3.3 of my paper below) if ZF is consistent, then there is a model of ZF + $(2)$ + $\lnot$ AC, which shows that over ZF the implication $(1) \Rightarrow (2)$ does not reverse.
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> The status of the above failure of reversibility over ZFC (as opposed to ZF), as well as the reversibility-status of $(2) \Rightarrow(3)$ over ZF or ZFC, were posed in a 2004-paper of mine (see Conjecture 4.3.2 of the paper below), and remain open.
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> 
> 


**More detail:** my paper [The Leibniz-Myscielski Axiom in Set Theory](http://academic2.american.edu/~enayat/LM.pdf) (Fund. Math. 181, 2004, pp.215-231) shows that *the existence of an injection (equivalently: a bijection) of V into P(Ord) is equivalent to a "logical" principal dubbed LM (for Leibniz-Mycieslki), where:* 


LM:$\ \ \ \forall x\forall y$ $[x\neq y\rightarrow\exists\alpha>\max
\{\rho(x),\rho(y)\}$ $Th(V\_{\alpha},\in,x)\neq Th(V\_{\alpha},\in,y)].$


In the above, $\rho(x)$ is the ordinal-valued set theoretic rank of $x$, and $Th(V\_{\alpha},\in,x)$ is the first order theory of the structure $(V\_{\alpha},\in,x)$. 


LM captures the spirit of Leibniz's principle on the identity of indiscernibles because, by a theorem of Mycielski, a consistent T completion of ZF includes LM iff T has a $\cal{M}$ model with no indisceribles , i.e., if $a$ and $b$ are a pair of distinct elements of $\cal{M}$, then there is a formula $\phi(x)$ in the language of set theory such that $\cal{M}\models\phi(a)\land\lnot\phi(b)$. 


As shown in the aforementioned paper, the existence of an injection from V into P(Ord) is also equivalent with the global version of the Kinna-Wagner Selection Principle, in other words, models of ZF + LM are precisely models of ZF in which there is a definable class function $\mathbf{F}$ that "chooses" a proper nonempty subset from each set with at least two elements, i.e., $\forall x(\left| x\right| \geq2\rightarrow \emptyset\neq\mathbf{F}(x)$ $\subsetneq x);$


