SUO: Re: Examples! Examples! Examples!
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EEE. Note 18
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I'm still waiting for the flash or insight that would show me
the structure of (2^L(!TLC!), c), the lattice of theories over
the TLC lexicon !TLC! = {a_1, ..., a_25}, and help me to see at
one glance just how the TLC axiom !a! acts to modulate its form.
In the meantime, let's go back to the more concrete images
of the lattice of models (2^TLC, c), that extends from the
null set {} to the full set TLC = <|a_1, ..., a_25|>, and
its isomorph the lattice of propositions (TLC^, =>), that
stretches from the constant function 0 : TLC -> B to the
constant function 1 : TLC -> B.
o-------------------------------------o-------------------------------------o
| | |
| TLC | |
| | |
| <|a_1, ..., a_25|> | 1 : TLC -> B |
| | |
| {<0000000000000000000000000>, | {<0000000000000000000000000> ~> 1, |
| <0000000000000000000000001>, | <0000000000000000000000001> ~> 1, |
| ... | ... |
| <1111111111111111111111110>, | <1111111111111111111111110> ~> 1, |
| <1111111111111111111111111>} | <1111111111111111111111111> ~> 1} |
| | |
| o | o |
| / \ | / \ |
| / \ | / \ |
| / \ | / \ |
| / \ | / \ |
| / 2^TLC \ | / TLC^ \ |
| / \ | / \ |
| / Pow (TLC) \ | / (TLC -> B) \ |
| ... ... | ... ... |
| \ 2^(2^25) / | \ 2^(2^25) / |
| \ subsets / | \ functions / |
| \ / | \ / |
| \ / | \ / |
| \ / | \ / |
| \ / | \ / |
| \ / | \ / |
| o | o |
| | |
| { } | ( ) |
| | |
| | 0 : TLC -> B |
| | |
| | {<0000000000000000000000000> ~> 0, |
| | <0000000000000000000000001> ~> 0, |
| | ... |
| | <1111111111111111111111110> ~> 0, |
| | <1111111111111111111111111> ~> 0} |
| | |
o-------------------------------------o-------------------------------------o
Figure 4. Subset Lattice Pow(TLC) and Proposition Lattice TLC^ = (TLC -> B)
Imagine that that I point an e-cam or web-cam through the window by my desk,
having sent off to a secret address in Schenectady for a complete set of 25
TLC detectors through which to filter the input. The various entities that
pass or that rest within the scope of my window, the bluebirds nesting in
the crabapple tree, the mourning doves in the spruce, the whole motley
crew at the bird feeder, the pedestrians and the bikers and the cars,
the chipmunks, rabbits, and squirrels, the grass, the holly bushes,
the rhododendron, the occasional crow, or duck, or canadian goose,
the mailbox with our house number on it, or maybe it's more like
a "numeral", or even just a plastic token thereof -- all these
things, in all their variety, get codified as so many vectors
of 25 bits. Now we all know that you can't play 20 questions
with Nature and win, but apparently 5 more are trump enough.
Still, if I now view Nature as speaking to me in code,
beaming me elements of B^25 until I can sift Her plan,
I eventually come to notice that it's apparently some
species of error-correcting code, in other words, the
codes that I receive over the longest time I know are
but a sparse selection of all the codes that might be.
To be continued ...
Jon Awbrey
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