I found the exact error noticed in Error in my theorem post.

The error was that I claimed that infimum of a greater set is greater (while in reality it’s lesser).

I will delete the erroneous theorem from my book soon.

It seems that there is an error in proof of this theorem.

Alleged counter-example:

$f=\bot$ and $z(p)=\top$ for infinite sets $A$ and $B$.

I am now attempting to locate the error in the proof.

I have proved (and added to my online book) the following theorem:

Theorem Let $f \in \mathsf{FCD} (A ; B)$ and $z \in \mathscr{F} (B)^A$. Then there is an (obviously unique) funcoid $g \in \mathsf{FCD} (A ; B)$ such that $\langle g\rangle x = \langle f\rangle x$ for nontrivial ultrafilters $x$ and $\langle g\rangle @\{ p \} = z (p)$ for $p \in A$

After I started to prove it, it took about a hour or like this to finish the proof.

I proved the following (in)equalities, solving my open problem which stood for a few months:

$\lvert \mathbb{R} \rvert_{>} \sqsubset \lvert \mathbb{R} \rvert_{\geq} \sqcap \mathord{>}$

$\lvert \mathbb{R} \rvert_{>} = \lvert \mathbb{R} \rvert_{>} \sqcap \mathord{>}$

The proof is currently available in the section “Some inequalities” of this PDF file.

Note that earlier I put online some erroneous proof related to this.

I have published What is physical reality? blog post in my other blog. The post is philosophical.

I proved that $\lvert \mathbb{R} \rvert_{\geq} \neq \lvert \mathbb{R} \rvert \sqcap \geq$ and so disproved one of my conjectures.

The proof is currently available in the section “Some inequalities” of this PDF file.

The proof isn’t yet thoroughly checked for errors.

Note that I have not yet proved $\lvert \mathbb{R} \rvert_{>} \neq \lvert \mathbb{R} \rvert \sqcap >$, but the proof is expected to be similar to the above.

I’ve moved the section “Some (example) values” to my main book file (instead of the draft file addons.pdf where it was previously).