The ultimate model is the world we live in….all of it, including us. There are no shortcuts…no map to crumble up and throw on the table so that we can cry,” aha….a fixed point!”
…except in the mind of the one making the claim. Show me the map and I’ll show you the holes.
The idea I’m after is whether “R” can be considered as the limiting case of an infinite Boolean structure…and that there may be a non-isomorphic model of the real numbers (not the usual set-theoretical one) that better fits “the real world”.
Start with the Stern-Brocot (S-B) tree…
https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree
https://www.cut-the-knot.org/blue/Stern.shtml
Continued Fractions | Sacred Geometry
If we count in binary in the right half of the S-B tree starting with 1 at the vertex labeled 2/1 and continue counting from left to right, row after row the result at each vertex gives a run length coding of the continued fraction representation for the real numbers on the right side of the tree (each vertex is the prefix of an infinite string…the “leaves” of the tree). The labels in rows on the left are complements of their counterparts on the right and likewise give a run length code for the continued fraction form of real numbers.
We can see that there are a series of interlocking relationships between the mediants of the S-B tree, the indices of the continued fraction form of real numbers, convergents of continued fractions of real numbers, counting vertices in binary, and the dyadic monoid related to a subgroup of the modular group.
I believe that there is an alternative model other than the usual interpretation of set theory that better “fits” the real world, one in which we view “random” as a lack of information…knowledge….not the lack of a “real world”…that the real world is not just a reflection of some “real” Platonic set-theoretical universe “out there”.
21st Century Paradox 