User:Emlightened/not-so-secret future blog posts/Chained Hydra

In case you're wondering, I hope to actually post this by the end of the month, as opposed to the other one which appears to be in stasis until we can approximate it with Rathjen's notation.

The notation is split into four stages: Stage 1 is equivalent to the Kirby-Parris Hydra (cutting from the right), Stage 2 is bounded at \(\psi(\Omega_\omega)\), Stage 3 terminates, and Stage 4 probably terminates.

I also hope to produce a wormlike variant of each stage. Note that each stage does not change any of the previous rules, but instead builds on them, without adding new syntax.

The basic syntax consists of '{', '}' and '[k]', where k is an integer.

Stage 1
On step k, if the rightmost node is the root node, delete it. Then, duplicate the parent node k times within the grandparent node.

Equivalently:

@{} --> @ ...{@{} '}'n --> ...'{@}'k '}'n

Where each @ is balanced, each expression is balanced, and ... is anything that doesn't violate this.

After doing each step, increment k.

Stage 2
WIP. You know that because there's nothing else here.