Meanwhile, I am currently writing a book called Ten Proofs of Gödel Incompleteness, which will include all these and many more proofs of the incompleteness theorem and closely related results. I plan to serialize this on my substack at Infinitely More.

@DanTuretsky I'm not quite following you. (But also, for James: I am thinking that the problem also makes perfect sense in Cantor space, where it seems even more natural. Probably equivalent though.)

I happened to have a look at Freiling's paper, and the arguments I give here are just what Freiling offers.

Smullyan has a whole book on these kinds of generalized fixed points, which covers the kind of case in which you are interested. So your question would seem to be part of that theory.

@Stanleysun I recommend that you use far more words in your question. Write complete English sentences to explain your ideas, with embedded formalism within those sentences, when necessary, for precise reference. For example: "Let us define the function $f$ by $f(x)=\ulcorner x\urcorner$, where $x$ will be any syntactic item in the language." (If that is indeed what you mean.) And that definition shouldn't be part of the defining property of $\mu$ but rather prior to it.

@AndrejBauer Oh! He was defining the function $f$, and then saying that $f$ is in $\mu$. Totally didn't get that.

My proposal amounts to: given a list of sequences $a_0,a_1,\ldots$ and a given computable sequence $b$, can we enumerate a sequence that is $b$, if $b$ does not occur amongst the $a_n$, but otherwise is some new sequence $b'$, if it does? If so, we could answer your question affirmatively. But unfortunately, I now think that this proposal isn't possible, since we can make a bad $a_n$ sequence by the Kleene recursion theorem that pretends to put $b$ as $a_0$ and waits until a non-$b$ bit appears in the answer, but then prevents $b$ from appearing. So my idea is not helpful.

Tree grows downward, since the empty sequence corresponds to 1 in the Boolean algebra. The partial order has no 0, but the branches stretch down towards 0 in the BA.

Sorry, I still don't get it. What do the brackets mean for you? Are these Gödel numbers? If $\mu$ is a set of functions, then those numbers are constant functions? Or what? And what do you mean by $f(x)=\ulcorner x\urcorner$, where neither $f$ nor $x$ has been quantified? Is $f(x)$ a function, or the value of the function at $x$ or a term? But $\ulcorner x\urcorner$ is the Gödel number of the variable symbol $x$? Or what? I am unsure of almost every single formal statement in your question.

I think of it as: take the tree $2^{<\omega_1}$ of all countable well ordered binary sequences as an order under extension, and consider the lower-cone topology (basic open = all extensions of a fixed element), and then take the regular open subsets of this space. The tree is the partial order approach to this forcing, the completion is the Boolean algebra approach. The tree is obviously countably closed, and it is dense in the Boolean algebra, which is therefore countably distributive.

Can't we arrange a path through $\mathcal{O}$ by specifying a cofinal $\omega$-sequence, such that the $n$th element of it is a simple limit in $\mathcal{O}$ that will ensure that a particular computable sequence is hit? That is, I want to reduce from the question of a path to the question of showing that a particular sequence arises at a given trivial kind of limit. Then, we can put them together to make a path on which every computable sequence will arise.

You say that $\mu$ is a set of functions, but then make an assertion about a certain Gödel code being an element of $\mu$. I'm not sure what you are trying to say without further explanation.