Rice's Theorem Proof using the Recursion Theorem?! - Easy Theory

Here we prove Rice's Theorem using the Recursion Theorem (https://www.youtube.com/watch?v=iXpp5X6WPkE&ab_channel=EasyTheory). The idea is very simple: have the machine obtain its own description, and then simulate itself based on whether or not it has the property, but with a "different" machine that has the opposite property. Specifically, if A has the property, and B does not, then if we find our own machine does have the property, we simulate B on the same input, not A (and simulate A if we do not have the property). This yields a contradiction about the property itself being decidable.

What is Rice's Theorem? It is a result that shows a lot of languages are undecidable using simple criteria, involving whether the language is nontrivial (not empty and not everything), and if the language involves Turing Machines and the criteria to be in the language is based on that of the TMs. See https://www.youtube.com/watch?v=kr7n_3LpWhc&ab_channel=EasyTheory for more details.

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