Systems of Logic Based on Ordinals
Systems of Logic Based on Ordinals was the PhD dissertation of the mathematician Alan Turing.[1][2]
Turing's thesis is not about a new type of
The thesis is an exploration of formal mathematical systems after Gödel's theorem. Gödel showed that for any formal system S powerful enough to represent arithmetic, there is a theorem G that is true but the system is unable to prove. G could be added as an additional axiom to the system in place of a proof. However this would create a new system S' with its own unprovable true theorem G', and so on. Turing's thesis looks at what happens if you simply iterate this process repeatedly, generating an infinite set of new axioms to add to the original theory, and even goes one step further in using transfinite recursion to go "past infinity", yielding a set of new theories Gα, one for each ordinal number α.
The thesis was completed at
References
- ProQuest 301792588.
- .
- ISBN 3-211-82637-8page 111
- ISBN 88-470-0320-2 pages 63-66 [1]
External links
- https://rauterberg.employee.id.tue.nl/lecturenotes/DDM110%20CAS/Turing/Turing-1939%20Sysyems%20of%20logic%20based%20on%20ordinals.pdf
- https://www.dcc.fc.up.pt/~acm/turing-phd.pdf
- https://web.archive.org/web/20121023103503/https://webspace.princeton.edu/users/jedwards/Turing%20Centennial%202012/Mudd%20Archive%20files/12285_AC100_Turing_1938.pdf
- "Turing's Princeton Dissertation". Princeton University Press. Retrieved January 10, 2012.
- Solomon Feferman (November 2006), "Turing's Thesis" (PDF), Notices of the AMS, 53 (10)