"Digital Physics" (the movie) is now free on Amazon Prime. It is also available for rent/purchase on iTunes and Vimeo, in case you are kind enough to support an independent filmmaker:) Come explore the world of self-reference, infinity, complexity, Gödel & Turing, and psychedelics with Khatchig!!
If you're interested in the Axiom of Choice, Godel, finitism, pseudo-randomness, complexity, information, and other foundational topics, check out the indie film "Digital Physics" on iTunes, Amazon, or Vimeo. Free packs of trading cards (with gum!) are available too! Check the website.
Let Mathematicians frolic in the Platonic world if they wish, but should physicists be using continuous math if our universe ends up by being discrete and finite?
"Say you are playing a game that needs you to pick a real number. If you choose a computable real number, you lose the game. If you choose a real number that is not computable, which the majority real numbers are, then you win. You can imagine yourself choosing a non-computable real number, and winning the game, if you build in the axiom of choice. But in the real world version of the game you will never have enough time or space to non-ambiguously specify this infinitely precise real number which has an infinite amount of non-compressible information (see Kolmogorov complexity)."-Khatchig, from "Digital Physics" (the movie)
Your talk of punishment vs rehabilitation is spot on, and very relevant to the "Doesn't the whole justice system collapse if there is no free will?" question. Have you ever heard of how some South American prisons are treating inmates with Ayahuasca, in order to rehabilitate their mind? Very interesting stuff considering the success rate.
I think in the end we will find out that both interpretations of whether free will exists might be correct, but with determinism being more fundamental. Think about a deterministic machine learning algorithm with a fixed input of training data. The agent/algorithm learns by reacting to it's data input/environment, has goals in the form of a cost function it is actively trying to minimize, has options to adjust its neural weights depending on what it experiences, yet we know the system including the environment is predetermined from the moment we kick off the program. -Khatchig, from "Digital Physics" (the movie).
I agree. Even though the axiom of choice is independent of ZF, I don't think it is self-evident axiom. It actually seems pretty counter-intuitive if you believe in infinite-precision real numbers that have an infinite amount of information and can't be compressed. I have more to say on this in "Digital Physics" (the movie). -Khatchig