Mid october 2020 notes from phone
I read about something simpler and more mathematically basic than set theory but I do not know what it is called, noticing most 2020 computers besides analog computers and quantum computers run off set theory ( and, or, not, nand etc), it seems possible
that CPUs and programs made from deeper math than set theory could have valuable characteristics and that higher level language computer programs compiled into deeper math than set theory hardware instructions could have new advantages, one way to test
this is a 2020 set theory emulation of a DM computer (DMC) and software to find out if any of it, including programs compiled into DM form are better in some ways, one possible way is that different DM CPU layouts might be based on fewer transistors or
even be made of semiconductors or other things that are actually not transistors at all, people could write simple software for the DM machine emulator then genetic algorithms (even cloud based 10,000 computer GA running a DMC (deep math computer)
emulation could improve the simple programs (add, subtract, multiply, divide, other math functions, Conway's game of life, perhaps even a simple jr high level math equation solver as well as any kind of program humans think a DMC machine would be good at,
disassembling the genetic algorithm optimized programs would teach human computer hardware engineers something and they could build DMCs in semiconductors (or other materials), if DMCs work better than set theory logic at the speed or also fewer-
instructions per data type (like floats that have less volume of pixel span (bytes) to do the same thing and calculate faster, then that is an advantage, DMCs and their programs also might be more fault tolerant and have different race conditions or even
no race conditions at all, different predictive branching and, at the hardware level different amounts of electrons utilized (energy consumption to do a thing) along with standalone DMCs there could be a DMC section of the CPU (this might be called a
core) so humans and compilers could take advantage of the things DMCs are better than what set theory computers do, also, photonic DMCs might be able to do very different things than set theory computers, instantly
The thing I read about in wikipedia did not, as far as I remember, have any of these names but there are similarities: "depends on what you take mathematics to be, and what your standards of proof are.
Do you allow, or expect, infinitary, potentially non-constructive reasoning in mathematics? If yes to the latter, do you accept quantifier-free transfinite induction up to epsilon-zero? Do you take Peano Arithmetic as your foundation? Then your
mathematics is consistent (Gentzen’s theorem).
If you take strictly finitary reasoning but include mathematical induction (and not transfinite induction), then Godel’s results leave it an open question, because there’s a statement true in the standard model that you can’t prove (Godel’s first
theorem) and you can’t use reasoning within the system to prove consistency (Second theorem), so that means either you must find a reduction strategy (say to Heyting Arithemetic, which then shifts the question there, and Godel showed this could be done)
or be comfortable with reasoning in a stronger theory, like a metatheory (which is typically seen as not favorable to the formalist or even constructivist positions).
Do you insist on Peano Arithmetic or Heyting Arithmetic as your foundation? There’s a result of Friedman and Meyer which says relevant arithmetic is consistent. But most logicians don’t like relevant logics, mostly because there’s so many of them,
they are not characterized by finite models, and the proof theory for these systems reflects principles of carrying out mathematics that most mathematicians might not like (for instance: reductio proof is out, because one must have sharing of variables
in a derivation). There’s also Presburger Arithmetic, which is decidable, complete, and consistent, and has induction, but only addition axioms. Is that a suitable foundation for mathematics? There’s also relative consistency proofs, and proofs of
consistency using primitive recursion and finite types but, you know, these are still not quite what we want, which is an absolute claim about consistency for foundations (whatever that might be).
For my money, I think the answer to the consistency of foundations question can be resolved by changing our assumptions about what the underlying logic should be — and very little has been done by way of work in substructural approaches to foundations.
Given Friedman and Meyer’s result, if absolute consistency is something independently valued, we should probably look elsewhere than Peano arithmetic or Is-mathematics-self-consistent
https://www.quora.com/Is-mathematics-self-consistent
Quantum dot millifiore ic lithography, imagine there could be a quantum dot that absorbs EUV at 230 nanometers, but is like a frequency doubling or tripling crystal and reemits light around 74 nanometers, much smaller than the EUV used to make computer
chips, then you naturally draw lines and circles with the quantum dots to make chip features three times smaller than EUV's 7 nanometer (your chips are better at 2.33 nanometer feature size) size, so, since the quantum dots are so tiny how do you draw
lines and circles with them, one way is to attach them to a polymer ladder or trirope (or structide).

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