161–170 of 271 posts
https://papers.ssrn.com/abstract=3603021
The article linked above gives a correct proof of inferential incompleteness.
Cf. Douglas Hofstadter's Metamagical Themas : > In March of 1977, I met the great AI pioneer Marvin Minsky for the first time. It was an unforgettable experience. One of the most memorable remarks he made to me was this one: "Gödel should just have thought up Lisp; it would have made the proof of his theorem much easier." I knew exactly what Minsky meant by that, I could see a grain of truth in it, and moreover I kne…
[1] https://www.goodreads.com/book/show/123471.I_Am_a_Strange_Lo...
Lispers learned marketing, bravo! The only issue is the missing credit to the Quanta article https://www.quantamagazine.org/how-godels-incompleteness-the... .
That's pretty rude of you. I didn't read that essay, but I did reference about 4 essays / books that I did read :). Went through the quanta article -- it seems reasonably different (doesn't go deep on how proof, subst, work, etc) -- I suspect the similarities stem from both of us reading Nagel and Newman's book.
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"I will just mention my main two quibbles" -- Poking through your history, why do you quibble on the word truth so often? Do we not agree there are statements that are true? Do we not agree there are provable true statements ? If there are unprovable statements in any consistent set of axioms, might we also conclude there also be an unprovable but true statements?
It's because it's a huge can of worms that leads to really grandiose claims that aren't supported by Godel's statements, of the sort the article is already beginning to make. It also shifts the conversation into becoming fundamentally a philosophical question rather than a logical or mathematical one, which is okay, but considerably changes the table stakes of what background knowledge we need. > If there are unprova…
In any case, my contribution here is something else: The most interesting part of the natural numbers object and its role in incomplete logical systems is to me the more humble notion of... unique prime factorisation.
If that had failed, so too would the Godel numbering system. I am not a number theorist, but this to me would be an interesting spin off from OP's post. The divisibility lattice is a good starting point, but apart from such basic constructions, the actual proof of unique prime factorisation to me is more historic than informative, and steps such as Bezout's Identity are just as "mythical" to me as Godel's Incompleteness Theorem, if we are using bad word choices. Conversely, numbers are just permutations on prime generators, but crucially, usually we start with addition before we start with multiplication.
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There is an extraordinarily minor typo: “PM-LIsp” I point it out only because your work here is so good that it feels wrong not to smooth out any small splinters. Thank you for making this!
Oi, thanks for the kind words and the catch! Updated (should take a few minutes to show up)
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> often by exaggerating their consequences. So you assert, many think differently. Edit: To clarify, are you asserting that they do /not/ extend into philosophy? What exaggerations are you specifically referring to?
Godel's theorems have philosophical ramifications, but then again so do basically all theorems given the right framing. It is nonetheless true that Godel's theorems are a particular focal point of discussion around mathematical philosophy (but again one which has surprisingly few ramifications for mathematics as a whole) and a very fruitful one at that. I'm not dismissing the idea of having a philosophical discussion…
I can't say I've followed all of the nuance in your (argument?) comments. I do know I'm going to be chewing on this for a while. I've come to grips with horror of saccades, fascinating to learn about new ways my mind, uh, decides what is true.
In any case, I appreciate your long thoughtful responses. May not mean much to you, but for at least one reader, it means a lot to me.
Very interesting. I'm wondering if this has implications in physics ? Since "the book of nature is written in mathematical language", does this mean that attempts to find a unified theory of everything are doomed ? I'm thinking not necessarily, since we wouldn't need to prove all possible physical phenomena, only those that actually happen, but how do we know if they are among the one that can be proven ?
This article spectacularly misses a big chunk of the point of Lisp, by applying Gödel-numbering to the printed representation of the syntax, rather than to the object that it denotes.