Timeline for What is Gödel's pairing function on ordinals?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Nov 12, 2012 at 10:22 | comment | added | Rodrigo Freire | Andreas: I think you got it right: According to secondary sources (Drake's book, page 132) the construction of the paper of 1939 is based on definability while the presentation in the monograph is based on Godel's operations. | |
| Nov 12, 2012 at 2:11 | comment | added | Andreas Blass | I think Rodrigo is probably right. A pairing function like this is needed when one develops constructibility using the Gödel functions (as opposed to the nowadays more common development using definability). And if I remember correctly, Gödel's book used the Gödel functions while his paper in the Proceedings of the National Academy used definability (or did I get that backward?). | |
| Nov 11, 2012 at 22:07 | comment | added | Rodrigo Freire | I have not checked the original sources, but I guess that Godel's pairing function is the inverse of this function described by Joel Hamkins. This inverse have a direct description in Shoenfield's Mathematical Logic, page 251. This (inverse) function is used by Shoenfield in the definition of the constructible model. One place to look is Godel's book on constructible sets and the consistency of GCH. | |
| Nov 11, 2012 at 16:56 | comment | added | Asaf Karagila♦ | Okay, according to Jech Set Theory historical notes the ordering is due to Hessenberg (from his book - which I couldn't find - "Grundbegriffe der Mengenlehre", 1906). | |
| Nov 11, 2012 at 16:32 | comment | added | Joel David Hamkins | Maybe the ordinal pairing functions are called Gödel coding not because Gödel invented this particular ordinal pairing function, but rather just because it is analogous to Gödel coding of sequences? | |
| Nov 11, 2012 at 16:26 | comment | added | Asaf Karagila♦ | I think that this coding is how Zermelo proved that $\aleph_\alpha\times\aleph_\alpha=\aleph_\alpha$. I'm not sure whether or not it was his discovery or someone else's and it can probably be checked in his 1904 paper. | |
| Nov 11, 2012 at 16:17 | vote | accept | Colin McLarty | ||
| Nov 11, 2012 at 16:15 | comment | added | Colin McLarty | Thanks. The absoluteness is just the kind of thing I wanted to check from the definition. | |
| Nov 11, 2012 at 16:14 | vote | accept | Colin McLarty | ||
| Nov 11, 2012 at 16:14 | |||||
| Nov 11, 2012 at 15:51 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Nov 11, 2012 at 15:41 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |