Gentzen's consistency proof initiated the program of ordinal analysis in proof theory. In this program, formal theories of arithmetic or set theory are assigned ordinal numbers that measure the consistency strength of the theories. A theory will be unable to prove the consistency of another theory with a higher proof theoretic ordinal.
gentzen's consistency proof initiated the program of ordinal analysis in proof theory in this program formal theories of arithmetic or set theory are assigned ordinal numbers that measure the consistency strength of the theories a theory will be unable to prove the consistency of another theory with a higher proof theoretic ordinal