Justin Attempts to Understand Part of “The Fabric of Reality”, Part 3

Summary so far of my attempts to understand David Deutsch’s discussion of Godel’s Theorem in The Fabric of Reality:

So in Part 1 we considered a type of “diagonal argument” applied to DD’s discussion of Cantgotu enviroments. Basically, DD said that a program for a VR generator has to have a discrete, quantized set of values for any variables, expressible as a finite sequence of symbols, and that it can be executed in a sequence of steps. Suppose we have an infinite set of possible VR programs meeting this criteria that some VR generator can run. We could define a second infinite set of programs (the Cantgotu list) in relation to the first by saying that this second set consists of programs whose characteristic is that, in any given minute, the VR environment they produce behaves differently than a VR environment in the first set. So in the first minute, Cantgotu Environment 1 behaves differently than plain old Environment 1, and in the second minute, Cantogu Environment 1 behaves differently than plain old Environment 2, and so on. So ultimately the Cantgotu Environment varies from every environment in the first list, and so this shows we can only run a fraction of logically possible environments.

In Part 2 we talked about the problem situation Godel’s Theorem was facing. Basically, mathematicians liked to believe that their proofs were on absolutely certain foundations. And Godel made this impossible by showing that 1) any set of rules of inference that could validate arithmetic proofs couldn’t validate its own consistency, and 2) a consistent set of rules of inference will fail to show as valid methods of proof which are in fact valid. And DD tells us that Godel used something like the diagonal arg that Cantor used to show there’s bigger infinities than natural numbers and that DD uses when talking about Cantgotu environments.

Basically, with regards to point 2, Godel considered some set of rules for making inferences, then showed how to construct a proposition you couldn’t either prove or disprove under the rules of inference. Then he showed the proposition was in fact true.

I feel like I get the general point but am not sure I could apply explain or apply this idea very well without knowing more technical details.

BTW I watched this video on Cantor’s diagonalization argument which I thought was okay (I tried a few different vids out). I looked for additional material (besides The Fabric of Reality) directly on Godel’s Theorem but it all seemed like it would be above my level of mathematical understanding.

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