About this proof, quoting from a talk by John W. Dawson Jr., an internationally recognized authority on the life and work of Kurt Gödel:
"And a final example of this sort of unfashionable pursuit is Gödel's
proof of the existence of God, which is a formalization in modal logic
of an argument — traces all the way back to Anselm — but in particular
an argument due to Leibniz. Gödel was very much interested in
Leibniz's philosophy, and so he felt that it would be possible to
formalize some of Leibniz's ideas. But because this was a theological
subject he was very cautious about this. Said very little about it
during his lifetime. And although word of it sort of leaked out, it
was only after his death that the details were really made public.
"In fact, when I was cataloging Gödel's papers, I realized that there was some concern among the triumvirate of mathematicians that were in charge of my work [...] that there might be some things in Gödel's Nachlass that needed to be restricted, that scholars would not be allowed access to. And it quickly became apparent that this was the primary thing they were thinking of. And it took some persuasion on my part to convince them
that while mathematicians may think it's silly to try to prove that God
exists, philosophers certainly don't, and that this was something
of real interest in philosophy. And so, eventually, unlike
Russell's papers, where — as you may know — large segments are
embargoed [...] you can see everything in Gödel's Nachlass, there aren't any restrictions on that."
From a quick skimming online, it sounds like there's a policy to embargo private correspondence until all involved haven't been around for at least 5 years.
In this case, it seems that Russell took a much younger woman as his third wife: [Patricia Russell, Countess Russell](https://en.wikipedia.org/wiki/Patricia_Russell,_Countess_Rus...). Wikipedia lists her as passing in 2004, so presumably there was an embargo on their correspondence until 2009.
About this proof, quoting from a talk by John W. Dawson Jr., an internationally recognized authority on the life and work of Kurt Gödel:
"And a final example of this sort of unfashionable pursuit is Gödel's proof of the existence of God, which is a formalization in modal logic of an argument — traces all the way back to Anselm — but in particular an argument due to Leibniz. Gödel was very much interested in Leibniz's philosophy, and so he felt that it would be possible to formalize some of Leibniz's ideas. But because this was a theological subject he was very cautious about this. Said very little about it during his lifetime. And although word of it sort of leaked out, it was only after his death that the details were really made public.
"In fact, when I was cataloging Gödel's papers, I realized that there was some concern among the triumvirate of mathematicians that were in charge of my work [...] that there might be some things in Gödel's Nachlass that needed to be restricted, that scholars would not be allowed access to. And it quickly became apparent that this was the primary thing they were thinking of. And it took some persuasion on my part to convince them that while mathematicians may think it's silly to try to prove that God exists, philosophers certainly don't, and that this was something of real interest in philosophy. And so, eventually, unlike Russell's papers, where — as you may know — large segments are embargoed [...] you can see everything in Gödel's Nachlass, there aren't any restrictions on that."
Source: https://youtu.be/4vpTFZdyjKM?t=2179