{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T17:25:20Z","timestamp":1774977920461,"version":"3.50.1"},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2022,4,25]],"date-time":"2022-04-25T00:00:00Z","timestamp":1650844800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In providing a good foundation for mathematics, set theorists often aim to develop the strongest theories possible and avoid those theories that place undue restrictions on the capacity to possess strength. For example, adding a measurable cardinal to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000181_inline1.png\"\/><jats:tex-math>\n$ZFC$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is thought to give a stronger theory than adding <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000181_inline2.png\"\/><jats:tex-math>\n$V=L$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and the latter is thought to be more restrictive than the former. The two main proponents of this style of account are Penelope Maddy and John Steel. In this paper, I\u2019ll offer a third account that is intended to provide a simple analysis of restrictiveness based on the algebraic concept of retraction in the category of theories. I will also deliver some results and arguments that suggest some plausible alternative approaches to analyzing restrictiveness do not live up to their intuitive motivation.<\/jats:p>","DOI":"10.1017\/s1755020322000181","type":"journal-article","created":{"date-parts":[[2022,4,25]],"date-time":"2022-04-25T10:53:14Z","timestamp":1650883994000},"page":"67-105","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["WHAT IS A RESTRICTIVE THEORY?"],"prefix":"10.1017","volume":"17","author":[{"given":"TOBY","family":"MEADOWS","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2022,4,25]]},"reference":[{"key":"S1755020322000181_r19","doi-asserted-by":"publisher","DOI":"10.2178\/bsl.1901010"},{"key":"S1755020322000181_r22","volume-title":"Interpreting G\u00f6del: Critical Essays","author":"Steel","year":"2014"},{"key":"S1755020322000181_r25","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-016-2341-5"},{"key":"S1755020322000181_r4","first-page":"401","article-title":"Does mathematics need new axioms?","volume":"6","author":"Feferman","year":"1999","journal-title":"American Mathematical Monthly"},{"key":"S1755020322000181_r9","unstructured":"[9] Hamkins, J. D. , & Seabold, D. E. (2012). Well-founded Boolean ultrapowers as large cardinal embeddings. Preprint, arXiv:1206.6075v1."},{"key":"S1755020322000181_r11","volume-title":"Set Theory","author":"Jech","year":"2003"},{"key":"S1755020322000181_r24","doi-asserted-by":"crossref","unstructured":"[24] Visser, A. (2006). Categories of theories and interpretations, Logic in Tehran. In Proceedings of the Workshop and Conference on Logic, Algebra and Arithmetic, Held October 18\u201322, pp. 284\u2013341.","DOI":"10.1201\/9781439865873-16"},{"key":"S1755020322000181_r3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511910616.004"},{"key":"S1755020322000181_r10","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020316000058"},{"key":"S1755020322000181_r26","doi-asserted-by":"publisher","DOI":"10.1017\/bsl.2016.34"},{"key":"S1755020322000181_r12","volume-title":"The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings","author":"Kanamori","year":"2003"},{"key":"S1755020322000181_r23","doi-asserted-by":"publisher","DOI":"10.1017\/9781316716892"},{"key":"S1755020322000181_r20","unstructured":"[20] Schatz, J. (2019). Axiom Selection and Maximize: Forcing Axioms vs. V = Ultimate L. PhD dissertation, University of California, Irvine."},{"key":"S1755020322000181_r6","volume-title":"Handbook of Set Theory","author":"Foreman","year":"2009"},{"key":"S1755020322000181_r17","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(72)90006-0"},{"key":"S1755020322000181_r2","unstructured":"[2] Enayat, A. (2016). Variations on a Visserian theme, In van Eijk J., Iemhoff R., and Joosten J., editors. Liber Amicorum Alberti, a Tribute to Albert Visser. London: College Publications, pp. 99\u2013110."},{"key":"S1755020322000181_r16","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1994-1224594-7"},{"key":"S1755020322000181_r18","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(74)90032-2"},{"key":"S1755020322000181_r1","unstructured":"[1] Aczel, P. (1988). Non-Well-Founded Sets. CSLI Lecture Notes. Stanford: Stanford University."},{"key":"S1755020322000181_r21","first-page":"115","volume-title":"More on the axiom of extensionality, in Essays on the foundations of mathematics, dedicated to Prof. A. H. 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Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}