I have been considering a (definability-free) weak form of the constructibility axiom, which is intended to capture the coarse structure of the constructible hierarchy. This means that this weak form is intended to capture the ordinal pattern abstracted from the constructible hierarchy if the fine behavior of the ordinal values of the constructible rank is covered up. It turns out that basic consequences of the constructibility axiom remain valid in the resulting theory:


Let $ZF\_{\rho}^-$ be the following theory: Its language has, in addition to $\in$, an unary function symbol $\rho$, and the axioms include all axioms of $ZF^-$ ($ZF$ minus the axiom of foundation) along with replacement and separation axioms for formulas containing $\rho$.


Now, add to $ZF\_{\rho}^-$ the following axiom: 


The function symbol $\rho$ is such that


1. $\forall x, \rho(x)$ is an ordinal.
2. $\forall \alpha, (\rho(\alpha)=\alpha)$.
3. $\forall x, y, (x\in y\rightarrow \rho(x)<\rho(y))$. (i.e., $\rho$ preserves membership.)
4. $\forall \alpha\: \exists f;(f:\alpha \cup \omega\rightarrow\left\{x:\rho(x)<\alpha\right\}$ is surjective). (i.e., $|\left\{x:\rho(x)<\alpha\right\}|\leq|\alpha\cup \omega|.)$
5. For every set $x$, $(i)$ if $x\in V\_{\omega}$, then $\rho(x)=rk(x)$, and $(ii)$ if $x\notin V\_{\omega}$, then given a transitive set $T$ containing $x$ and $r:T\rightarrow T$ satisfying 1-4 above, $\rho(x)<r(x)^+$.


A function symbol satisfying 1-5 is called a *minimal ordinal connection* in $V$. It is minimal with respect to cardinality. The constructible rank is a minimal ordinal connection in $L$ (the minimality clause 5 is a consequence of Godel's condensation lemma). In this sense, the above axiom is a weak form of the constructibility axiom. (If $a\subseteq\omega$, then the $L[a]$-rank is also a minimal ordinal connection in $L[a]$.)


The axiom of foundation, the axiom of choice and $GCH$ are theorems of the above theory. Also, if $\kappa$ is inaccessible, then $\left\{x:\rho(x)<\kappa\right\}=V\_{\kappa}$. These are basic consequences of $ZF^- + V=L$ (if $\rho$ is interpreted as the constructible rank).


Now, it seems that the usual proofs of $V\neq L$ from, for example, the existence of a measurable cardinal will not work here, because they involve definability and absoluteness aspects of $L$ which are not present in this case. I have two questions:


* Is this weak form of $V=L$ consistent with measurable cardinals?
* Is this weak form of $V=L$ inconsistent with large cardinal notions assumed to be consistent with $ZFC$?



Is this weak form of $V=L$ (in)consistent with large cardinals?
This is consistent with very large cardinals. It follows from the work of Friedman and Holy, who showed that an abstract version of local club condensation and acceptability are simultaneously consistent with the existence of very large cardinals. See their papers:


(1) Condensation and Large Cardinals.
Fundamenta Mathematicae 215, no. 2, pp 133-166, 2011.


(2) A Quasi Lower Bound on the Consistency Strength of PFA.
Transactions of the AMS 366, pp 4021-4065, 2014.


