The Lambda Point

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

AK = Antti Karttunen

AK: So far I haven't found/constructed any ranking scheme
    which would be practical to implement as a computer
    algorithm, in a sense that the sequence {max integer
    mapped to the size n tree} wouldn't grow too steeply.
    I reckon this is also the problem with Matula/Göbel
    encoding of rooted unoriented trees.

Ok, there are lots and lots of mappings between any two countable sets,
so I guess that the "interesting" ones are relative to one's interest,
one's end-in-view, and most likely one's personal sense of aesthetics.
I was initially interested in the correspondences that might serve as
mathematical "labor-saving devices", allowing us to transfer theorems,
what we know and can prove, among different domains, especially among
algebra, geometry, and logic -- what one of my teachers used to call
the "lambda point", after the triple critical point of transitions
among gas, liquid, and solid phases.

 From this perspective, it's not a problem, may even be a feature,
that some relatively big numbers have some interesting aspects
of their structure captured by some relatively small graphs.

I haven't said anything about the logical side of this yet,
but remember that it was folks like Gödel, not to mention
Peirce, who lobbed us into this particular briar patch.

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o