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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

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    $\begingroup$ 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. $\endgroup$
    – Asaf Karagila ♦
    Commented Dec 11, 2023 at 18:04
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    $\begingroup$ @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. $\endgroup$
    – Joel David Hamkins
    Commented Dec 11, 2023 at 18:26
  • $\begingroup$ @JoelDavidHamkins precisely. $\endgroup$
    – Lorenzo
    Commented Dec 11, 2023 at 18:52
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    $\begingroup$ 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 $L$. $\endgroup$
    – Miha Habič
    Commented Dec 11, 2023 at 19:21

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Yes, there is indeed such a paper, see Mack Stanley's paper "Coding a generic extension of L".

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    $\begingroup$ Theorem 1 in that paper states, "Relative to the consistency of ZFC, over L it is consistent that...". But what does "...over L it is consistent that..." mean? I think he just means "it is consistent that" and drop the "over L" bit. $\endgroup$
    – Joel David Hamkins
    Commented Dec 12, 2023 at 3:13
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    $\begingroup$ He simply means the ground model is L, and forces over it. $\endgroup$
    – Mohammad Golshani
    Commented Dec 12, 2023 at 7:40
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    $\begingroup$ Yes, I know that is what he wants to mean. But he should be making a consistency assertion about ZFC and L, not an assertion in L about consistency. I have noticed that students often make the same kind of mistake in their dealings with metatheory, mixing up consistency assertions with substantive statements of the background set theory. For example, you might find someone saying things like: "Assume Con(ZFC+kappa is measurable cardinal). Then there is a forcing extension where kappa has many normal measures." It is just wrongly stated in my view. $\endgroup$
    – Joel David Hamkins
    Commented Dec 12, 2023 at 14:20
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    $\begingroup$ In fact, in his case he doesn't need the consistency assumption at all. Rather, the theorem could be stated as a ZFC theorem: there is a definable tame class forcing notion forcing the existence of a real x that is not set-generic over L. No need for Con(ZFC) preliminary or Con(conclusion), since this simply follows from the forcing fact, which is the main result. $\endgroup$
    – Joel David Hamkins
    Commented Dec 12, 2023 at 14:31
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I don’t have enough reputation to comment. I just wanted to say that you might be interested in the following resources which avoid the morass of fine structure theory around this issue.

S.D. Friedman, Coding without Fine Structure.

S.D. Friedman, A Simpler Proof of Jensen's Coding Theorem.

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