Abstract
Assuming \(\text {ZF}+\text {DC}\), we prove that if there exists a strong partition cardinal greater than \(\varTheta \), then (1) there is an inner model of \(\text {ZF}+\text {AD}+\text {DC}+ {{{\mathbb {R}}} }^{{\#}}\) exists, and (2) there is an inner model of \(\text {ZF}+\text {AD}+\text {DC}+ (\exists \kappa >\varTheta )\,(\kappa \) is measurable). Here \(\varTheta \) is the supremum of the ordinals which are the surjective image of the set of reals \({{{\mathbb {R}}} }\).
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Cunningham, D.W. A strong partition cardinal above \(\varTheta \) . Arch. Math. Logic 56, 403–421 (2017). https://doi.org/10.1007/s00153-017-0529-8
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DOI: https://doi.org/10.1007/s00153-017-0529-8