One of the most incredible results in modern set theory, due to Jensen, is that given any model of $\sf ZFC$, there is a class forcing which adds a real number $r$ and in the extension $V=L[r]$. Moreover, if we assume some basic things like $\sf GCH$, then cardinals are not collapsed, so in particular things like inaccessible cardinals are preserved.


So in Jensen's result the "coding model" is $L$. But $L[x]$ is somewhat of a dull model. It doesn't have any sharps, measurable cardinals, or larger cardinals. Even those with reasonably canonical inner models.



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> **Question.** Given any reasonably canonical core model $K$, can we code the universe into a real *over $K$* while preserving large cardinals that are captured by $K$?
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In other words, can we code the universe into a real while preserving measurable cardinals? Can we code the universe into a real while preserving strong, Woodin, etc.?


And the obvious follow-up question, what is the bare minimum needed from an inner model $M$ to be a coding model?



Coding the universe into a real over better core models
For measurable cardinals, the answer is yes and is due to Sy Friedman. See [Coding Over a Measurable Cardinal](https://www.jstor.org/stable/2274810?origin=crossref&seq=1).


There is some difficulty to extend the result to the context of Woodin cardinals, see [Genericiy and large cardinals](https://www.worldscientific.com/doi/abs/10.1142/S0219061305000420).


For strong cardinals, this is open, though for some simple sets, say the collection of Prikry sequences for the Prikry product, the answer is yes, see [Killing the GCH everywhere with a single real](https://projecteuclid.org/journals/journal-of-symbolic-logic/volume-78/issue-3/Killing-the-GCH-everywhere-with-a-single-real/10.2178/jsl.7803060.short).


See also [Coding over core models](http://www.math.uni-bonn.de/people/david/files/fri-schi-schr-coding-K.pdf)


