God! I always love me a good Mphlee post. You've really progressed as a mathematician, and your referencing is highly fascinating.
I'd also like to add, as an aside, more than anything, that any negative rank in an analytic sense results in singular behaviour. Any attempt to write:
\[
a[s]b\\
\]
For \(a[0]b = a+b,\,a[1]b = ab,\,a[2]b = a^b\), while still satisfying goodstein's equation--results in:
\[
a[-1] b = a+1\\
\]
But \(s \approx -1\) displays strong singular behaviour. So when defining "analytic semi-operators", the negative integers act as singularities, in and of themself, where they look like successorship \(a[-2]b = a+1\), but are actually very weird singularities. Where as something like \(a [-1.5] b\) is plausible, and can be analytic, you won't see this at the negative integers themself.
And before we get into analytic semi-operators, I believe I have finalized the proof that Bennet's operations can be massaged to produce Goodstein's equation. Again though, I cannot run code (I've given up on programming it). But the math looks 100%, as the kids say.
I'd also like to add, as an aside, more than anything, that any negative rank in an analytic sense results in singular behaviour. Any attempt to write:
\[
a[s]b\\
\]
For \(a[0]b = a+b,\,a[1]b = ab,\,a[2]b = a^b\), while still satisfying goodstein's equation--results in:
\[
a[-1] b = a+1\\
\]
But \(s \approx -1\) displays strong singular behaviour. So when defining "analytic semi-operators", the negative integers act as singularities, in and of themself, where they look like successorship \(a[-2]b = a+1\), but are actually very weird singularities. Where as something like \(a [-1.5] b\) is plausible, and can be analytic, you won't see this at the negative integers themself.
And before we get into analytic semi-operators, I believe I have finalized the proof that Bennet's operations can be massaged to produce Goodstein's equation. Again though, I cannot run code (I've given up on programming it). But the math looks 100%, as the kids say.
