{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T12:10:51Z","timestamp":1759839051686},"reference-count":32,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2020,7,2]],"date-time":"2020-07-02T00:00:00Z","timestamp":1593648000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2021,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020320000143_inline1.png\" \/><jats:tex-math>\n$\\mathsf {DC}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-preserving symmetric submodels of forcing extensions. Hence,<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020320000143_inline2.png\" \/><jats:tex-math>\n$\\mathsf {ZF}+\\mathsf {DC}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020320000143_inline3.png\" \/><jats:tex-math>\n$\\mathsf {ZF}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020320000143_inline4.png\" \/><jats:tex-math>\n$\\mathsf {ZF}+\\mathsf {DC}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020320000143_inline5.png\" \/><jats:tex-math>\n$\\mathsf {ZFC}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Our results confirm<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020320000143_inline6.png\" \/><jats:tex-math>\n$\\mathsf {ZF} + \\mathsf {DC}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> as a natural foundation for a significant portion of \u201cclassical mathematics\u201d and provide support to the idea of this theory being also a natural foundation for a large part of set theory.<\/jats:p>","DOI":"10.1017\/s1755020320000143","type":"journal-article","created":{"date-parts":[[2020,7,2]],"date-time":"2020-07-02T08:56:01Z","timestamp":1593680161000},"page":"225-249","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":4,"title":["DEPENDENT CHOICE, PROPERNESS, AND GENERIC ABSOLUTENESS"],"prefix":"10.1017","volume":"14","author":[{"given":"DAVID","family":"ASPER\u00d3","sequence":"first","affiliation":[]},{"given":"ASAF","family":"KARAGILA","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,7,2]]},"reference":[{"key":"S1755020320000143_r3","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-86580-9.50024-0"},{"key":"S1755020320000143_r21","doi-asserted-by":"publisher","DOI":"10.2307\/2273318"},{"key":"S1755020320000143_r8","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-07-04297-3"},{"key":"S1755020320000143_r29","doi-asserted-by":"publisher","DOI":"10.1142\/S021906131000095X"},{"key":"S1755020320000143_r11","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/059"},{"key":"S1755020320000143_r19","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771612"},{"key":"S1755020320000143_r32","unstructured":"[32] Yoshinobu, Y . (2019). Properness under closed forcing. 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(2018). Incompatible category forcing axioms. Preprint. arXiv:1805.08732."},{"key":"S1755020320000143_r24","unstructured":"[24] Sch\u00f6lder, J. J. (2013). Forcing Axioms Through Iterations of Minimal Counterexamples. 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