“So what had Gödel done? Budiansky’s treatment of Gödel’s proof is light, and I would recommend the 1950s classic “Gödel’s Proof” by Ernest Nagel and James Newman for a semi-popular treatment. Even today Gödel’s seminal paper is comprehensible in its entirety only to specialists in the field. But in a nutshell, what Gödel had found using an ingenious bit of self-referential mapping between numbers and mathematical statements was that any consistent mathematical system that could support the basic axioms of arithmetic as described in Russell and Whitehead’s work would always contain statements that were unprovable. This ingenious scheme included a way of encoding mathematical statements as numbers, allowing numbers to “talk about themselves”. What was worse and even more fascinating was that the axiomatic system of arithmetic would contain statements that were true, but whose truth could not be proven using the axioms of the system – Gödel thus showed that there would always be a statement G in this system which would, like the old Liar’s Paradox, say, “G is unprovable”. If G is true it then becomes unprovable by definition, but if G is false, then it would be provable, thus contradicting itself. Thus, the system would always contain ‘truths’ that are undecidable within the framework of the system. And lest one thought that you could then just expand the system and prove those truths within that new system, Gödel infuriatingly showed that the new system would contain its own unprovable truths, and ad infinitum. This is called the First Incompleteness Theorem. – Ashutosh Jogalekar, “Kurt Gödel’s Open World” at 3quarksdaily (July 12, 2021)”
The big materialist project was then formally over. Many people did not realize that fact and, in any event, it would take a long time to mop up. We are still mopping.News, “How eccentric mathematician Kurt Gödel opened the world” at Mind Matters News
As often happens, few people understood the significance of what had just happened. The one exception was John von Neumann.
Takehome: The First Incompleteness Theorem means that the big materialists project is essentially already over. But it takes a long time to mop up.
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Gregory Chaitin’s “almost” meeting with Kurt Gödel: This hard-to-find anecdote gives some sense of the encouraging but eccentric math genius.