[FOM] Freeman Dyson on Inexhaustibility

[A.P. Hazen]

   Charles Silver's report of Freeman Dyson's comments remined me of 
the only "New  Yorker" magazine "Talk of the town" column ever 
reviewed by Alonzo Church in the J.S.L. (XVII 289(6) -- the column 
appeared in the August 23, 1952 issue of the magazine).
   Gödel had been awarded an honorary degree, and the citation 
included the  intriguing comment that his work was incomprehensible 
to most people.  When the "New Yorker" staff writer interviewed him 
about this, Gödel admitted that his Continuum Hypothesis work was a 
bit technical,  but asserted that the Incompleteness Theorem was 
quite accessible.  His one-line summary of the significance of the 
incompleteness  results was that they showed that mathematics was 
"inexhaustible."
--
   As for the snide comments of computer scientists, who Karlis 
Podnieks  reports as claiming that the Incompleteness result  is a 
simple corollary of the more fundamental theorem that there are 
non-recursive r.e. sets... There is **some** truth to this (despite 
its inversion of the historical sequence of discovery).  Since Turing 
Machines and their behavior are (via Gödel-numbering) DESCRIBABLE in 
the language of Arithmetic, the First Incompleteness Theorem -- the 
existence of unprovable truths -- follows easily from the 
Unsolvability of the Halting Problem.  Because of the intuitive 
nature of Turing  Machines (and because this argument avoids the 
puzzling "self-referential"  sentence), I have found this the easiest 
proof of First Incompleteness to present to students (toward  the end 
of the semester, when time is short).
   The Second Incompleteness Theorem, on the other hand....
--
Allen Hazen
Philosophy Department
University of Melbourne