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We prove that it is relatively consistent with the usual axioms ofmathematics that all automorphisms of the Calkin algebra are inner. Togetherwith a 2006 Phillips--Weaver construction of an outer automorphism using theContinuum Hypothesis, this gives a complete solution to a 1977 problem ofBrown-Douglas-Fillmore. We also give a simpler and self-contained proof of thePhillips--Weaver result.

All automorphisms of the Calkin algebra are inner