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Abstract For which choices of$$X,Y,Z\in \{\Sigma ^1_1,\Pi ^1_1\}$$does no sufficiently strongX-sound andY-definable extension theory prove its ownZ-soundness? We give a complete answer, thereby delimiting the generalizations of Gödel’s second incompleteness theorem that hold within second-order arithmetic. 

Abstract Goldschmidtite is a new perovskite-group mineral (IMA No. 2018-034) with the ideal formula (K,REE,Sr)(Nb,Cr)O3. A single grain of goldschmidtite with a maximum dimension of ∼100 μm was found as an inclusion in a diamond from the Koffiefontein pipe in South Africa. In addition to the dark green and opaque goldschmidtite, the diamond contained a Cr-rich augite (websteritic paragenesis) and an intergrowth of chromite, Mg-silicate, and unidentified K-Sr-REE-Nb-oxide. Geothermobarometry of the augite indicates that the depth of formation was ∼170 km. The chemical composition of gold-schmidtite determined by electron microprobe analysis (n = 11, WDS, wt%) is: Nb2O5 44.82, TiO2 0.44, ThO2 0.10, Al2O3 0.35, Cr2O3 7.07, La2O3 11.85, Ce2O3 6.18, Fe2O3 1.96, MgO 0.70, CaO 0.04, SrO 6.67, BaO 6.82, K2O 11.53, total 98.53. The empirical formula (expressed to two decimal places) is (K0.50La0.15Sr0.13Ba0.09Ce0.08)Σ0.95(Nb0.70Cr0.19Fe0.05Al0.01Mg0.04Ti0.01)Σ1.00O3. Goldschmidtite is cubic, space group Pm3m, with unit-cell parameters: a = 3.9876(1) Å, V = 63.404(6) Å3, Z = 1, resulting in a calculated density of 5.32(3) g/cm3. Goldschmidtite is the K-analog of isolueshite, (Na,La)NbO3. Raman spectra of goldschmidtite exhibit many second-order broad bands at 100 to 700 cm–1 as well as a pronounced peak at 815 cm–1, which is possibly a result of local ordering of Nb and Cr at the B site. The name goldschmidtite is in honor of the eminent geochemist Victor Moritz Goldschmidt (1888–1947), who formalized perovskite crystal chemistry and identified KNbO3 as a perovskite-structured compound.