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Gilles
Gilles
2:34 PM
> there is one possible "weak" precedent for this. Godels thm and diagonalization may have been loosely based on Richards paradox which was from literary work. but note it took extremely advanced mathematicians to convert it into legitimate mathematical statements/properties. – vzn
> @vzn: the very Wikipedia page you link to dates Richard's Paradox to 1905; diagonalization dates back to 1891. So Richard's Paradox is likely based on diagonalization, not the other way around. – Niel de Beaudrap
> obviously you are referring to cantors 1891 proof, good point. however (my understanding) note it was highly controversial and not taken seriously by math community at time of publication. also (my understanding) a cantor-like proof of TM undecidability (countable vs uncountable sets) was discovered later than the TM proof (not sure by who). — vzn
> saw the richards paradox vs diagonalization argument comparison in some published work on math history but dont recall which one right now. note cantor page also cites russell's paradox.... – vzn
> also, my understanding, it is possibly only modern theory that recognizes godels proof as a "diagonalization" argument, that connection (as with the TM proof) was not apparent at the time it was 1st proved. overall all some questions with subtle historical nuance... – vzn
> @vzn: Regardless of whether Cantor's argument was taken seriously in 1891, this does not mean that Richard's Paradox was not informed by an understanding of diagonalization by 1905. – Niel de Beaudrap
> But in short, you are mostly just hypothesizing that Godel's work may have been influenced by Richard's, both which you stipulate somehow as being independent of Cantor, without having any evidence yourself that Richard's paradox was not influenced by Cantor, or that Godel was influenced by Richard. In what way are those remarks meant to be useful? – Niel de Beaudrap
(manually migrated from comments on cs.stackexchange.com/questions/7726/…)
 
vzn
vzn
3:15 PM
niel you are putting words into my mouth. am saying [from the start] its not perfectly clear what role richards paradox played in the development of Godels thm. others have noted the parallels. the wikipedia page mentions Paul Finsler writing to Godel about richards paradox in 1931. Godel wrote back saying the Finsler work was insufficiently formalized.
acc to wikipedia Finsler came up with something in 1926 on the subj.
as for "diagonalization", the modern understanding has evolved substantially over time.
dont know when it was 1st realized that Godels proof is actually (like) a "diagonalization" but dont think that was the awareness at the time of publication. so when I originally referred to "diagonalization", was referring to the modern advanced understanding.
 
 
8 hours later…
vzn
vzn
10:53 PM
[it looks like gilles moving these comments lost the hyperlinks. so will hyperlink again...]
Richard's paradox
In logic, Richard's paradox is a semantical antinomy in set theory and natural language first described by the French mathematician Jules Richard in 1905. Today, the paradox is ordinarily used in order to motivate the importance of carefully distinguishing between mathematics and metamathematics. The paradox was also a motivation in the development of predicative mathematics. Description The original statement of the paradox, due to Richard (1905), has a relation to Cantor's diagonal argument on the uncountability of the set of real numbers. The paradox begins with the observation tha...
Cantor's diagonal argument
Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began. The diagonal argument was not Cantor's first proof of the uncountability of the real numbers; it was actually published much later than his first...
 

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