Talk:2610: Assigning Numbers
Does this imply that Gödel's Incompleteness Theorem isn't correct? And that it's method is bunk? Please help! -Seer 162.158.107.230 02:08, 23 April 2022 (UTC) I believe the intention is that the theorem is not part of the set of bad data science, just that they share this one feature.
Isn't the Gödel number for a theorem calculated by multiplying the numbers of the components together, so complicated theorems would have larger numbers? If so, the current explanation that this isn't a good way to judge fields is wrong. I'm not too sure though. MrCandela (talk) 05:52, 23 April 2022 (UTC)
- I do not believe that the title suggests renumbering theorems with Gödel numbers, but averaging the existing theorem numbers. Or otherwise, MrCandela's suggestion would be the way to go: Complicated Theorems have larger numbers. Sebastian --172.68.110.133 08:10, 23 April 2022 (UTC)
- Yeah a quick look at some magazines like this one and I think Randall has a point MrCandela (talk) 09:48, 23 April 2022 (UTC)
I wish I'd started the explanation off when I first saw it (somone posted the first Transcript whilst I was pondering, so I left off). I think there's some serious re-editing to be done, but basically it points to someone (Cueball, a dabbling armchair mathematician faced with some not directly mathematically-based problem) thinking that 'all' it takes is to encode the whatever-it-is, arbitrarily, and then with a few easy equations something useful cannbe derived. When, in reality, even if this is possible (ignoring the "takes the age of the universe to permute things to find the right answer" sort of sticking-block) it depends upon a good numerical encoding (enough attention to detail, but not too much, and in the right sort of way) and possibly quite a lot of data-demunging and filtration (again, just the right amount and in the correct manner) to pop out the "answer" being looked for. For some things, this can be easy, though there are always statistical pitfalls/etc. For others ("life, the universe and everything", say) the task is far more complex and the result ("42"?) might not seem to be a very useful result for various reasons. And, on top this, there's Gödel. But that's an additional punchline, not the whole scope of the original joke. ...Anyway, this long comment is why I held back from writing the original Explanation, but I might yet wrangle my thoughts into what's since been put there. While trying not to tread upon too many toes and alternate explanations. Which is the hardest bit, I think... 172.70.86.64 15:48, 23 April 2022 (UTC)
Just a comment about the technicalities of Gödel's First Incompleteness Theorem: The 'third' possibility presented here misunderstands the term 'true but unprovable'. When mathematicians say 'true but unprovable' in the context of Gödel's Incompleteness Theorems, what they mean is 'true in the standard model but unprovable in the formal system'. The Gödel sentence is certainly true for the standard natural numbers, by contradiction: assume that the Gödel sentence is false for the standard naturals, which means that there exists a standard natural number which is the Gödel number for the proof of the Gödel sentence. Then we could decode the Gödel number into a proof (of the formal system) proving the Gödel sentence true; a contradiction. (Note that the preceding proof by contradiction can be formalised in ZFC, but not in the formal system under study.) The reason why the Gödel sentence is unprovable in the formal system is because, from the point of view of the formal system, there might be a non-standard natural number which is the Gödel number for the proof of the Gödel sentence (and non-standard numbers cannot be decoded into a proof); or there might not be. --Underbase (talk) 04:56, 24 April 2022 (UTC)
- Regarding this, I know that the policy on this site is to include every possible interpretation, but the page mentioned is an html page (and not a pdf) that was not peer reviewed (thus not recognized by the community), and as mentioned by the user above it fails understand the concepts it is talking about. I do not think this site should be spreading this kind of idea. I believe Randall Monroe himself would be against this.
- I also believe the current explanation is both incorrect about explaining the seeming paradox of the Gödel conjecture, & therefore somewhat incorrect about this joke. It is surely the transition from abstract to quantized - the act of applying limited formal numbering to potentially unbounded or otherwise non-standard terms - which incurs incompleteness? Within the constraints of a formal system of standard natural numbers, true≠provable, & therein lies the internal (but not total) contradiction. That's the contradiction, right? & the joke is that numbering theorems by their complexity, is not generally a productive approach for 'doing math' on them, in any sense but an abstract analytical one?
- ProphetZarquon (talk) 17:54, 24 April 2022 (UTC)
- I do not believe the Title Text calls for "calculating the average of all the fields' theorems' Gödel numbers". It asks for 'the lowest average theorem number'. The average of all, is not the average of each. The Title Text wants the average of each of the fields' theorems' Gödel numbers.
- ProphetZarquon (talk) 17:54, 24 April 2022 (UTC)
Paradoxicality argument
I think that revision 231000 should be removed. My explanation of what's wrong with the linked site is as follows:
Up until the section "Gödel's String", nothing is incorrect. Furthermore, the first wrong line is numbered (49), and says that Gödel's statement is equivalent to "This statement is not a theorem (of any formal system)." This is where he goes wrong, for writing down a formula for "n proves m" requires inclusion of the formal system in which this proof happens. As such, the correct translation of Gödel's statement is "This statement is not a theorem of [system]", which it indeed is not. Then he says that "We have decided that Gödel's string cannot be a theorem and neither can its negation" (true, after Rosser's trick) and therefore that this gives us "~<G∨~G>" (which is false). He has commited the sin of confusing truth and provability here.
His discussion of the Epimenides string ("This statement is not true") is accurate, except for the claim that the truth predicate is "as valid an extension to [PA] as [the provability and quining] extensions were". This is false. The provability and quining predicates can be constructed in PA and thus are not "extensions" so much as "shorthand"; this was Gödel's contribution: to see that PA can talk about provability of statements in any fixed formal system. The truth predicate is not definable in PA, as he quite ably proves (suppose it was definable, then you could write down the Epimenides sentence in PA, and thereby prove false).
The section "Gödel's Error" is just plain silly.172.70.114.147 19:28, 24 April 2022 (UTC)
What if we just change it to say something along the lines of "Certain logical systems allow values to be 'not false' without being necessarily 'true'; Godel's theorem is based on an axiomatic assumption that every statement is either true or false."?108.162.221.163 06:06, 25 April 2022 (UTC)