**Systems of Logic Based on Ordinals** was the PhD dissertation of the mathematician Alan Turing.

The thesis is an exploration of formal mathematical systems after Gödel's theorem. Gödel showed for that any formal system S powerful enough to represent arithmetic, there is a theorem G which 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 considers iterating the process to infinity, creating a system with an infinite set of axioms.

The thesis was completed at Princeton under Alonzo Church and was a classic work in mathematics which introduced the concept of ordinal logic.

Martin Davis states that although Turing's use of a computing oracle is not a major focus of the dissertation, it has proven to be highly influential in theoretical computer science, e.g. in the polynomial time hierarchy.