# On the existence of a real which is not set-generic over $L$

Recall that a real $$r$$ is set-generic over $$L$$ if there is a constructible forcing notion $$\mathbb{P}$$ and some $$L$$-generic filter $$G\subset\mathbb{P}$$ such that $$r \in L[G]$$.
I know that Jensen's coding the universe is a (very powerful and complicated) method to produce (class-generic) reals that are not set-generic over $$L$$. This is the only method I've heard of to produce such reals, but I also feel that invoking Jensen's coding to show the consistency, relative to $$\mathsf{ZF}$$, of a non-set-generic-over-$$L$$ real should be overkill. So my question is:

• Is there an "easy" (that is, easier than Jensen's coding or, at least, that uses only a relatively small fragment of the overall coding machinery) proof of the consistency, relative to $$\mathsf{ZF}$$, of $$\mathsf{ZFC}+$$"There is a real which is not set-generic over $$L$$"?

Thanks

• Something needs to be clarified here. Just a real which is not set generic? Take any real which codes a generic for $\operatorname{Col}(\omega,V)$ and that's not going to be set generic. Why is there one? Well, why is there one which codes the universe in the Jensen case? We're working over set-models (otherwise working in $V=L$ there's no real which is not generic for the trivial forcing over $L$), so those set-models might as well be countable. Dec 11, 2023 at 18:04
• @Asaf, the OP wants a model of ZFC that has a real that is not set generic over $L$. This is expressible in the language of set theory. Your argument about $\text{Col}(\omega,V)$ does not provide such a model, since that extension will not have ZFC. The coding the universe argument does provide such a model, but he is seeking a simpler direct argument. Dec 11, 2023 at 18:26
• @JoelDavidHamkins precisely. Dec 11, 2023 at 18:52
• I have this memory that one can build "fake" versions of $0^\sharp$ over countable models without many hypotheses at all. What I mean by this is start with a countable model $M$ (might as well have it satisfy $V=L$ too) and build a real $x$ coding a premouse iterable up to $\omega_1^M$ (but not fully iterable in $V$). If this works then $M[x]$ should see that $x$ is not set-generic over $L$. Dec 11, 2023 at 19:21