Oh, author is friends with Sebastian Marshall, that explains a thing or two :).
What Gödel Discovered
181–190 of 271 posts
Re: What Gödel Discovered
#182So if we have a theory expressive enough to make statements about ordinary (Peano) arithmetic, we can always form a self-referential statement within the framework of this theory which we can not prove or disprove. So far, so good. Here is my question: What happens if we restrict/weaken the theory to preclude self-referential statements? Obviously, we will lose our ability to express certain arithmetic statements which correspond to self-referential statements in the original theory. But what else? Is that the only class of statements we lose? Also, are there any other kinds of statements that still make the theory incomplete?
Re: What Gödel Discovered
#183This is remarkably well done. I'm reading Hofstadter's Godel, Escher, Bach and found this at the perfect time. Thanks for the write up!
Man, I just searched about that book, and it is like there is a world I don't know anything about. Would you mind sharing the names of your favorite books?
I assume you might be looking for books of a similar (technical) flavour, of which I don't have too many to recommend, I'm afraid. However, here's some (across different genres) that are in my memory at this moment:
Finance/statistics : The Black Swan by Nicholas Nassim Taleb The Drunkards Walk by Leonard Mlodinow
Math/science history : Euclid's Window by Leonard Mlodinow A Short History of Nearly Everything by Bill Bryson
Physics: Newton's Principia for the Common Reader by S. Chandrasekhar
Lit: The Brothers Karamazov by Fyodor Dostoevsky House of Leaves by Mark Z. Danielewski East of Eden by John Steinbeck
Philosophy: Zen Mind, Beginner's Mind by Shunryu Suzuki Any of the upanishads but probably Kena Upanishad, Isha Upanishad, or Prashna Upanishad at first (selected for (relative) ease in readership by yours truly) Zen and the Art of Motorcycle Maintenance by Robert M. Pirsig (for a gentle introduction into Eastern thought)
I'm missing countless others but this is what I have right now. Thanks for the prompt and happy reading! :)
Re: What Gödel Discovered
#184Earlier quoted context omitted.
Man, I just searched about that book, and it is like there is a world I don't know anything about. Would you mind sharing the names of your favorite books?
EGB is one of my favourite books ever, but I think it doesn't explain Gödel's theorem as simple as it could. Of course, there are just so many wonderful things in EGB that this is not a fatal flaw by any means. There is a fantastic book called "The Universal Computer: The Road from Leibniz to Turing" by Martin D. Davis that makes a very good job of explaining the context of Gödel's theorem, the theorem itself and the…
Yup. Also recommend to those interested in existential philosophy and artificial intelligence as well. It really is a terrific book!
Re: What Gödel Discovered
#185Earlier quoted context omitted.
> Alternatively do you believe that the real numbers are an artificial mathematical construction which can have different properties depending on which axioms you are willing to admit? Then there will be properties of the real numbers which are independent of ZFC as a result of Godel's incompleteness theorems. This is actually not correct. The propositions that are undecidable are NOT AT ALL connected to the proposit…
> The propositions that are undecidable are NOT AT ALL connected to the propositions that are independent. I'm not quite sure what you mean by undecidable propositions, but I assume it to mean something like the following? Propositions which must be true or must be false (i.e. are satisfied/not satisfied respectively by all models which satisfy the overall theory), but for which there is no finite proof that they hol…
> To take a step back, the assertion that there are fundamentally undecidable mathematical propositions in the real sense you're laying out (as opposed to more formal treatments of truth in model theory) is a statement of the Platonist school of mathematical philosophy.
To go one step forward, these undecidable/uncomputable statements are computed by humans all the time, which makes the Platonist school of mathematics much more than an intellectual curio.
Re: What Gödel Discovered
#186Just imagine the number of times the author has used the word "imagine", lol.
Re: What Gödel Discovered
#187Earlier quoted context omitted.
Thank you, glad you liked it :)
Well, I disliked it. I see proof that every valid formula can be converted into a Godel number. However, every invalid formula can also be converted into a Godel number. So, if we are able to construct a number, then it proves what?
The conversion of formulae to Gödel numbers is basically an implementation detail, the fact it can be done allows us to define theorems on the natural numbers that describe properties of the logical system. Encoding an invalid formula is of course possible, but I don't see why that would be a problem, it doesn't inherently 'prove' anything.
Re: What Gödel Discovered
#188> For example, a gentleman called Frege discovered that he could craft a theory of sets, which could represent just about everything. For numbers, for example, he could do something like this: [ 0 is {}, 1 is {{}}, 2 is { {{}} {} }, etc. ] I don’t know Frege too well, but believe this is due to von Neumann, not Frege: https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_def...
To me it read as though the author cleverly worded it as 'he could do something like this' to separate Frege's idea (theory of sets) from the specific example.
Re: What Gödel Discovered
#189I've read Godel, Escher, Bach, and I've read this. It's a very nice explanation.
But everywhere I see, Godel's theorem is touted as some kind of deep philosophical insight, whereas from what I understand, informally it could be rephrased as "if you have a usable language for mathematical proofs, some phrases in that language must be neither true nor false (i.e. nonsensical)".
Nonsensical phrases in our human languages in nothing new, so the conclusion becomes that much less exciting. Though I'm sure it's valuable for fundamental math theory.
Re: What Gödel Discovered
#190I've read Godel, Escher, Bach, and I've read this. It's a very nice explanation. But everywhere I see, Godel's theorem is touted as some kind of deep philosophical insight, whereas from what I understand, informally it could be rephrased as "if you have a usable language for mathematical proofs, some phrases in that language must be neither true nor false (i.e. nonsensical)". Nonsensical phrases in our human language…
To me reading this article it was striking to consider the relationship between language and (mathematicsl/logical) truth this way. In my mind it correlates with the quantum phenomenon that observing something fundamentally alters its state.