Recursively open doors

The following is Hofstadter-Gödel reasoning with my own speculative layer on top. (I don't care if I'm correct, but I do want to understand.)

tldr: Logic itself cannot be fully defined, but it can define itself perfectly well, and can be used to create definitions for logical destinations we hadn't thought of before. There's a logical asymptote in every system.

Longer version:

  1. Arranging formal components of logic (according to their own rules) always leads to valid logical statements. (Source: Principia Mathematica, via Hofstadter.)

  2. There are valid logical statements that cannot be proven by the components of logic at hand. (Source: Gödel's incompleteness theorems, via Hofstadter.)

  3. Arrangements of formal components of logic can themselves be validly arranged. let's call this these "hyper-logic components". we can get just as much logical flexibility from these as from the original, more-fundamental pieces. including, inevitably, the ability to construct a valid but unprovable hyper-logical statement. (This is my interpretation of Hofstadter.)

  4. Components of hyper logic can be recombined to form hyper-hyper-logic components, which afford more valid-but-unprovable arrangements. (This is my own speculative theory here.) In this fashion, we can look back to our original components of logic, and then our components of hyper-logic, then our new level of hyper-hyper-logic, and we can anticipate infinitely increasing levels beyond that. at every level, there's always an open door of unprovability. It's a logical asymptote, though. We can keep going on forever, and we'll always get closer (i.e. more logically precise), but we'll never reach the limit.

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